Mathematical Methods in Combinatorics

An Introduction to
Mathematical Methods in Combinatorics
Renzo Sprugnoli
Dipartimento di Sistemi e Informatica
Viale Morgagni, 65 - Firenze (Italy)
January 18, 2006
2
Contents
1 Introduction 5
1.1 What is the Analysis of an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The Analysis of Sequential Searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Binary Searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Closed Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 The Landau notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Special numbers 11
2.1 Mappings and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Counting permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Dispositions and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 The Pascal triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Harmonic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Walks, trees and Catalan numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.10 Stirling numbers of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.11 Stirling numbers of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.12 Bell and Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Formal power series 25
3.1 Definitions for formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The basic algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Formal Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Operations on formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Coefficient extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.8 Lagrange inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.9 Some examples of the LIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.10 Formal power series and the computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.11 The internal representation of expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.12 Basic operations of formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.13 Logarithm and exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Generating Functions 39
4.1 General Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Some Theorems on Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 More advanced results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Common Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 The Method of Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.7 Some special generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3
4 CONTENTS
4.8 Linear recurrences with constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.9 Linear recurrences with polynomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.10 The summing factor method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.11 The internal path length of binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.12 Height balanced binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.13 Some special recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Riordan Arrays 55
5.1 Definitions and basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 The algebraic structure of Riordan arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 The A-sequence for proper Riordan arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Simple binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Other Riordan arrays from binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.6 Binomial coefficients and the LIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.7 Coloured walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.8 Stirling numbers and Riordan arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.9 Identities involving the Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Formal methods 67
6.1 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Context-free languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Formal languages and programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.4 The symbolic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.5 The bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.6 The Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.7 The Difference Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.8 Shift and Difference Operators - Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.9 Shift and Difference Operators - Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.10 The Addition Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.11 Definite and Indefinite summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.12 Definite Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.13 The Euler-McLaurin Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.14 Applications of the Euler-McLaurin Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7 Asymptotics 83
7.1 The convergence of power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 The method of Darboux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 Singularities: poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Poles and asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.5 Algebraic and logarithmic singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.6 Subtracted singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.7 The asymptotic behavior of a trinomial square root . . . . . . . . . . . . . . . . . . . . . . . . 89
7.8 Hayman’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.9 Examples of Hayman’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8 Bibliography 93
Chapter 1
Introduction
1.1 What is the Analysis of an
Algorithm
An algorithm is a finite sequence of unambiguous
rules for solving a problem. Once the algorithm is
started with a particular input, it will always end,
obtaining the correct answer or output, which is the
solution to the given problem. An algorithm is real-
ized on a computer by means of a program, i.e., a set
of instructions which cause the computer to perform
the elaborations intended by the algorithm.
So, an algorithm is independent of any computer,
and, in fact, the word was used for a long time before
computers were invented. Leonardo Fibonacci called
“algorisms” the rules for performing the four basic
operations (addition, subtraction, multiplication and
division) with the newly introduced arabic digits and
the positional notation for numbers. “Euclid’s algo-
rithm” for evaluating the greatest common divisor of
two integer numbers was also well known before the
appearance of computers.
Many algorithms can exist which solve the same
problem. Some can be very skillful, others can be
very simple and straight-forward. A natural prob-
lem, in these cases, is to choose, if any, the best al-
gorithm, in order to realize it as a program on a par-
ticular computer. Simple algorithms are more easily
programmed, but they can be very slow or require
large amounts of computer memory. The problem
is therefore to have some means to judge the speed
and the quantity of computer resources a given algo-
rithm uses. The aim of the “Analysis of Algorithms”
is just to give the mathematical bases for describing
the behavior of algorithms, thus obtaining exact cri-
teria to compare different algorithms performing the
same task, or to see if an algorithm is sufficiently good
to be efficiently implemented on a computer.
Let us consider, as a simple example, the problem
of searching. Let S be a given set. In practical cases
S is the set N of natural numbers, or the set Z of
integer numbers, or the set R of real numbers or also
the set A
∗
of the words on some alphabet A. How-
ever, as a matter of fact, the nature of S is not essen-
tial, because we always deal with a suitable binary
representation of the elements in S on a computer,
and have therefore to be considered as “words” in
the computer memory. The “problem of searching”
is as follows: we are given a finite ordered subset
T = (a
1
, a
2
, . . . , a
n
) of S (usually called a table, its
elements referred to as keys), an element s ∈ S and
we wish to know whether s ∈ T or s ∈ T, and in the
former case which element a
k
in T it is.
Although the mathematical problem “s ∈ T or
not” has almost no relevance, the searching prob-
lem is basic in computer science and many algorithms
have been devised to make the process of searching
as fast as possible. Surely, the most straight-forward
algorithm is sequential searching: we begin by com-
paring s and a
1
and we are finished if they are equal.
Otherwise, we compare s and a
2
, and so on, until we
find an element a
k
= s or reach the end of T. In
the former case the search is successful and we have
determined the element in T equal to s. In the latter
case we are convinced that s ∈ T and the search is
unsuccessful.
The analysis of this (simple) algorithm consists in
finding one or more mathematical expressions de-
scribing in some way the number of operations per-
formed by the algorithm as a function of the number
n of the elements in T. This definition is intentionally
vague and the following points should be noted:
• an algorithm can present several aspects and
therefore may require several mathematical ex-
pressions to be fully understood. For example,
for what concerns the sequential search algo-
rithm, we are interested in what happens during
a successful or unsuccessful search. Besides, for
a successful search, we wish to know what the
worst case is (Worst Case Analysis) and what
the average case is (Average Case Analysis) with
respect to all the tables containing n elements or,
for a fixed table, with respect to the n elements
it contains;
• the operations performed by an algorithm can be
5
6 CHAPTER 1. INTRODUCTION
of many different kinds. In the example above
the only operations involved in the algorithm are
comparisons, so no doubt is possible. In other
algorithms we can have arithmetic or logical op-
erations. Sometimes we can also consider more
complex operations as square roots, list concate-
nation or reversal of words. Operations depend
on the nature of the algorithm, and we can de-
cide to consider as an “operation” also very com-
plicated manipulations (e.g., extracting a ran-
dom number or performing the differentiation of
a polynomial of a given degree). The important
point is that every instance of the “operation”
takes on about the same time or has the same
complexity. If this is not the case, we can give
different weights to different operations or to dif-
ferent instances of the same operation.
We observe explicitly that we never consider exe-
cution time as a possible parameter for the behavior
of an algorithm. As stated before, algorithms are
independent of any particular computer and should
never be confused with the programs realizing them.
Algorithms are only related to the “logic” used to
solve the problem; programs can depend on the abil-
ity of the programmer or on the characteristics of the
computer.
1.2 The Analysis of Sequential
Searching
The analysis of the sequential searching algorithm is
very simple. For a successful search, the worst case
analysis is immediate, since we have to perform n
comparisons if s = a
n
is the last element in T. More
interesting is the average case analysis, which intro-
duces the first mathematical device of algorithm anal-
ysis. To find the average number of comparisons in a
successful search we should sum the number of com-
parisons necessary to find any element in T, and then
divide by n. It is clear that if s = a
1
we only need
a single comparison; if s = a
2
we need two compar-
isons, and so on. Consequently, we have:
C
n
=
1
n
(1 + 2 + +n) =
1
n
n(n + 1)
2
=
n + 1
2
This result is intuitively clear but important, since
it shows in mathematical terms that the number of
comparisons performed by the algorithm (and hence
the time taken on a computer) grows linearly with
the dimension of T.
The concluding step of our analysis was the execu-
tion of a sum—a well-known sum in the present case.
This is typical of many algorithms and, as a matter of
fact, the ability in performing sums is an important
technical point for the analyst. A large part of our
efforts will be dedicated to this topic.
Let us now consider an unsuccessful search, for
which we only have an Average Case analysis. If U
k
denotes the number of comparisons necessary for a
table with k elements, we can determine U
n
in the
following way. We compare s with the first element
a
1
in T and obviously we find s = a
1
, so we should
go on with the table T
′
= T ` ¦a
1
¦, which contains
n −1 elements. Consequently, we have:
U
n
= 1 +U
n−1
This is a recurrence relation, that is an expression
relating the value of U
n
with other values of U
k
having
k < n. It is clear that if some value, e.g., U
0
or U
1
is known, then it is possible to find the value of U
n
,
for every n ∈ N. In our case, U
0
is the number of
comparisons necessary to find out that an element
s does not belong to a table containing no element.
Hence we have the initial condition U
0
= 0 and we
can unfold the preceding recurrence relation:
U
n
= 1 +U
n−1
= 1 + 1 +U
n−2
= =
= 1 + 1 + + 1
. .. .
ntimes
+U
0
= n
Recurrence relations are the other mathematical
device arising in algorithm analysis. In our example
the recurrence is easily transformed into a sum, but
as we shall see this is not always the case. In general
we have the problem of solving a recurrence, i.e., to
find an explicit expression for U
n
, starting with the
recurrence relation and the initial conditions. So, an-
other large part of our efforts will be dedicated to the
solution or recurrences.
1.3 Binary Searching
Another simple example of analysis can be performed
with the binary search algorithm. Let S be a given
ordered set. The ordering must be total, as the nu-
merical order in N, Z or R or the lexicographical
order in A
∗
. If T = (a
1
, a
2
, . . . , a
n
) is a finite or-
dered subset of S, i.e., a table, we can always imag-
ine that a
1
< a
2
< < a
n
and consider the fol-
lowing algorithm, called binary searching, to search
for an element s ∈ S in T. Let a
i
the median el-
ement in T, i.e., i = ⌊(n + 1)/2⌋, and compare it
with s. If s = a
i
then the search is successful;
otherwise, if s < a
i
, perform the same algorithm
on the subtable T
′
= (a
1
, a
2
, . . . , a
i−1
); if instead
s > a
i
perform the same algorithm on the subtable
T
′′
= (a
i+1
, a
i+2
, . . . , a
n
). If at any moment the ta-
ble on which we perform the search is reduced to the
empty set ∅, then the search is unsuccessful.
1.4. CLOSED FORMS 7
Let us consider first the Worst Case analysis of
this algorithm. For a successful search, the element
s is only found at the last step of the algorithm, i.e.,
when the subtable on which we search is reduced to
a single element. If B
n
is the number of comparisons
necessary to find s in a table T with n elements, we
have the recurrence:
B
n
= 1 +B
⌊n/2⌋
In fact, we observe that every step reduces the table
to ⌊n/2⌋ or to ⌊(n−1)/2⌋ elements. Since we are per-
forming a Worst Case analysis, we consider the worse
situation. The initial condition is B
1
= 1, relative to
the table (s), to which we should always reduce. The
recurrence is not so simple as in the case of sequential
searching, but we can simplify everything considering
a value of n of the form 2
k
−1. In fact, in such a case,
we have ⌊n/2⌋ = ⌊(n −1)/2⌋ = 2
k−1
−1, and the re-
currence takes on the form:
B
2
k
−1
= 1 +B
2
k−1
−1
or β
k
= 1 +β
k−1
if we write β
k
for B
2
k
−1
. As before, unfolding yields
β
k
= k, and returning to the B’s we find:
B
n
= log
2
(n + 1)
by our definition n = 2
k
−1. This is valid for every n
of the form 2
k
−1 and for the other values this is an
approximation, a rather good approximation, indeed,
because of the very slow growth of logarithms.
We observe explicitly that for n = 1, 000, 000, a
sequential search requires about 500,000 comparisons
on the average for a successful search, whereas binary
searching only requires log
2
(1, 000, 000) ≈ 20 com-
parisons. This accounts for the dramatic improve-
ment that binary searching operates on sequential
searching, and the analysis of algorithms provides a
mathematical proof of such a fact.
The Average Case analysis for successful searches
can be accomplished in the following way. There is
only one element that can be found with a single com-
parison: the median element in T. There are two
elements that can be found with two comparisons:
the median elements in T
′
and in T
′′
. Continuing
in the same way we find the average number A
n
of
comparisons as:
A
n
=
1
n
(1 + 2 + 2 + 3 + 3 + 3 + 3 + 4 +
+ (1 +⌊log
2
(n)⌋))
The value of this sum can be found explicitly, but the
method is rather difficult and we delay it until later
(see Section 4.7). When n = 2
k
− 1 the expression
simplifies:
A
2
k
−1
=
1
2
k
−1
k
¸
j=1
j2
j−1
=
k2
k
−2
k
+ 1
2
k
−1
This sum also is not immediate, but the reader can
check it by using mathematical induction. If we now
write k2
k
−2
k
+1 = k(2
k
−1) +k −(2
k
−1), we find:
A
n
= k +
k
n
−1 = log
2
(n + 1) −1 +
log
2
(n + 1)
n
which is only a little better than the worst case.
For unsuccessful searches, the analysis is now very
simple, since we have to proceed as in the Worst Case
analysis and at the last comparison we have a failure
instead of a success. Consequently, U
n
= B
n
.
1.4 Closed Forms
The sign “=” between two numerical expressions de-
notes their numerical equivalence as for example:
n
¸
k=0
k =
n(n + 1)
2
Although algebraically or numerically equivalent, two
expressions can be computationally quite different.
In the example, the left-hand expression requires n
sums to be evaluated, whereas the right-hand ex-
pression only requires a sum, a multiplication and
a halving. For n also moderately large (say n ≥ 5)
nobody would prefer computing the left-hand expres-
sion rather than the right-hand one. A computer
evaluates this latter expression in a few nanoseconds,
but can require some milliseconds to compute the for-
mer, if only n is greater than 10,000. The important
point is that the evaluation of the right-hand expres-
sion is independent of n, whilst the left-hand expres-
sion requires a number of operations growing linearly
with n.
A closed form expression is an expression, depend-
ing on some parameter n, the evaluation of which
does not require a number of operations depending
on n. Another example we have already found is:
n
¸
k=0
k2
k−1
= n2
n
−2
n
+ 1 = (n −1)2
n
+ 1
Again, the left-hand expression is not in closed form,
whereas the right-hand one is. We observe that
2
n
= 22 2 (n times) seems to require n−1 mul-
tiplications. In fact, however, 2
n
is a simple shift in
a binary computer and, more in general, every power
α
n
= exp(nln α) can be always computed with the
maximal accuracy allowed by a computer in constant
time, i.e., in a time independent of α and n. This
is because the two elementary functions exp(x) and
ln(x) have the nice property that their evaluation is
independent of their argument. The same property
holds true for the most common numerical functions,
8 CHAPTER 1. INTRODUCTION
as the trigonometric and hyperbolic functions, the Γ
and ψ functions (see below), and so on.
As we shall see, in algorithm analysis there appear
many kinds of “special” numbers. Most of them can
be reduced to the computation of some basic quan-
tities, which are considered to be in closed form, al-
though apparently they depend on some parameter
n. The three main quantities of this kind are the fac-
torial, the harmonic numbers and the binomial co-
efficients. In order to justify the previous sentence,
let us anticipate some definitions, which will be dis-
cussed in the next chapter, and give a more precise
presentation of the Γ and ψ functions.
The Γ-function is defined by a definite integral:
Γ(x) =

∞
0
t
x−1
e
−t
dt.
By integrating by parts, we obtain:
Γ(x + 1) =

∞
0
t
x
e
−t
dt =
=

−t
x
e
−t

∞
0
+

∞
0
xt
x−1
e
−t
dt
= xΓ(x)
which is a basic, recurrence property of the Γ-
function. It allows us to reduce the computation of
Γ(x) to the case 1 ≤ x ≤ 2. In this interval we can
use a polynomial approximation:
Γ(x + 1) = 1 +b
1
x +b
2
x
2
+ +b
8
x
8
+ǫ(x)
where:
b
1
= −0.577191652 b
5
= −0.756704078
b
2
= 0.988205891 b
6
= 0.482199394
b
3
= −0.897056937 b
7
= −0.193527818
b
4
= 0.918206857 b
8
= 0.035868343
The error is [ǫ(x)[ ≤ 3 10
−7
. Another method is to
use Stirling’s approximation:
Γ(x) = e
−x
x
x−0.5
√
2π

1 +
1
12x
+
1
288x
2
−
−
139
51840x
3
−
571
2488320x
4
+

.
Some special values of the Γ-function are directly
obtained from the definition. For example, when
x = 1 the integral simplifies and we immediately find
Γ(1) = 1. When x = 1/2 the definition implies:
Γ(1/2) =

∞
0
t
1/2−1
e
−t
dt =

∞
0
e
−t
√
t
dt.
By performing the substitution y =
√
t (t = y
2
and
dt = 2ydy), we have:
Γ(1/2) =

∞
0
e
−y
2
y
2ydy = 2

∞
0
e
−y
2
dy =
√
π
the last integral being the famous Gauss’ integral.
Finally, from the recurrence relation we obtain:
Γ(1/2) = Γ(1 −1/2) = −
1
2
Γ(−1/2)
and therefore:
Γ(−1/2) = −2Γ(1/2) = −2
√
π.
The Γ function is defined for every x ∈ C, except
when x is a negative integer, where the function goes
to infinity; the following approximation can be im-
portant:
Γ(−n +ǫ) ≈
(−1)
n
n!
1
ǫ
.
When we unfold the basic recurrence of the Γ-
function for x = n an integer, we find Γ(n + 1) =
n (n − 1) 2 1. The factorial Γ(n + 1) =
n! = 1 2 3 n seems to require n −2 multi-
plications. However, for n large it can be computed
by means of the Stirling’s formula, which is obtained
from the same formula for the Γ-function:
n! = Γ(n + 1) = nΓ(n) =
=
√
2πn

n
e

n

1 +
1
12n
+
1
288n
2
+

.
This requires only a fixed amount of operations to
reach the desired accuracy.
The function ψ(x), called ψ-function or digamma
function, is defined as the logarithmic derivative of
the Γ-function:
ψ(x) =
d
dx
ln Γ(x) =
Γ
′
(x)
Γ(x)
.
Obviously, we have:
ψ(x + 1) =
Γ
′
(x + 1)
Γ(x + 1)
=
d
dx
xΓ(x)
xΓ(x)
=
=
Γ(x) +xΓ
′
(x)
xΓ(x)
=
1
x
+ψ(x)
and this is a basic property of the digamma function.
By this formula we can always reduce the computa-
tion of ψ(x) to the case 1 ≤ x ≤ 2, where we can use
the approximation:
ψ(x) = ln x −
1
2x
−
1
12x
2
+
1
120x
4
−
1
252x
6
+ .
By the previous recurrence, we see that the digamma
function is related to the harmonic numbers H
n
=
1 + 1/2 + 1/3 + + 1/n. In fact, we have:
H
n
= ψ(n + 1) +γ
where γ = 0.57721566 . . . is the Mascheroni-Euler
constant. By using the approximation for ψ(x), we
1.5. THE LANDAU NOTATION 9
obtain an approximate formula for the Harmonic
numbers:
H
n
= ln n +γ +
1
2n
−
1
12n
2
+
which shows that the computation of H
n
does not
require n − 1 sums and n − 1 inversions as it can
appear from its definition.
Finally, the binomial coefficient:

n
k

=
n(n −1) (n −k + 1)
k!
=
=
n!
k!(n −k)!
=
=
Γ(n + 1)
Γ(k + 1)Γ(n −k + 1)
can be reduced to the computation of the Γ function
or can be approximated by using the Stirling’s for-
mula for factorials. The two methods are indeed the
same. We observe explicitly that the last expression
shows that binomial coefficients can be defined for
every n, k ∈ C, except that k cannot be a negative
integer number.
The reader can, as a very useful exercise, write
computer programs to realize the various functions
mentioned in the present section.
1.5 The Landau notation
To the mathematician Edmund Landau is ascribed
a special notation to describe the general behavior
of a function f(x) when x approaches some definite
value. We are mainly interested to the case x →
∞, but this should not be considered a restriction.
Landau notation is also known as O-notation (or big-
oh notation), because of the use of the letter O to
denote the desired behavior.
Let us consider functions f : N →R (i.e., sequences
of real numbers); given two functions f(n) and g(n),
we say that f(n) is O(g(n)), or that f(n) is in the
order of g(n), if and only if:
lim
n→∞
f(n)
g(n)
< ∞
In formulas we write f(n) = O(g(n)) or also f(n) ∼
g(n). Besides, if we have at the same time:
lim
n→∞
g(n)
f(n)
< ∞
we say that f(n) is in the same order as g(n) and
write f(n) = Θ(g(n)) or f(n) ≈ g(n).
It is easy to see that “∼” is an order relation be-
tween functions f : N →R, and that “≈” is an equiv-
alence relation. We observe explicitly that when f(n)
is in the same order as g(n), a constant K = 0 exists
such that:
lim
n→∞
f(n)
Kg(n)
= 1 or lim
n→∞
f(n)
g(n)
= K;
the constant K is very important and will often be
used.
Before making some important comments on Lan-
dau notation, we wish to introduce a last definition:
we say that f(n) is of smaller order than g(n) and
write f(n) = o(g(n)), iff:
lim
n→∞
f(n)
g(n)
= 0.
Obviously, this is in accordance with the previous
definitions, but the notation introduced (the small-
oh notation) is used rather frequently and should be
known.
If f(n) and g(n) describe the behavior of two al-
gorithms A and B solving the same problem, we
will say that A is asymptotically better than B iff
f(n) = o(g(n)); instead, the two algorithms are
asymptotically equivalent iff f(n) = Θ(g(n)). This is
rather clear, because when f(n) = o(g(n)) the num-
ber of operations performed by A is substantially less
than the number of operations performed by B. How-
ever, when f(n) = Θ(g(n)), the number of operations
is the same, except for a constant quantity K, which
remains the same as n → ∞. The constant K can
simply depend on the particular realization of the al-
gorithms A and B, and with two different implemen-
tations we may have K < 1 or K > 1. Therefore, in
general, when f(n) = Θ(g(n)) we cannot say which
algorithm is better, this depending on the particular
realization or on the particular computer on which
the algorithms are run. Obviously, if A and B are
both realized on the same computer and in the best
possible way, a value K < 1 tells us that algorithm
A is relatively better than B, and vice versa when
K > 1.
It is also possible to give an absolute evaluation
for the performance of a given algorithm A, whose
behavior is described by a sequence of values f(n).
This is done by comparing f(n) against an absolute
scale of values. The scale most frequently used con-
tains powers of n, logarithms and exponentials:
O(1) < O(ln n) < O(
√
n) < O(n) <
< O(nln n) < O(n
√
n) < O(n
2
) <
< O(n
5
) < < O(e
n
) <
< O(e
e
n
) < .
This scale reflects well-known properties: the loga-
rithm grows more slowly than any power n
ǫ
, how-
ever small ǫ, while e
n
grows faster than any power
10 CHAPTER 1. INTRODUCTION
n
k
, however large k, independent of n. Note that
n
n
= e
nln n
and therefore O(e
n
) < O(n
n
). As a mat-
ter of fact, the scale is not complete, as we obviously
have O(n
0.4
) < O(
√
n) and O(ln ln n) < O(ln n), but
the reader can easily fill in other values, which can
be of interest to him or her.
We can compare f(n) to the elements of the scale
and decide, for example, that f(n) = O(n); this
means that algorithm A performs at least as well as
any algorithm B whose behavior is described by a
function g(n) = Θ(n).
Let f(n) be a sequence describing the behavior of
an algorithm A. If f(n) = O(1), then the algorithm
A performs in a constant time, i.e., in a time indepen-
dent of n, the parameter used to evaluate the algo-
rithm behavior. Algorithms of this type are the best
possible algorithms, and they are especially interest-
ing when n becomes very large. If f(n) = O(n), the
algorithm A is said to be linear; if f(n) = O(ln n), A
is said to be logarithmic; if f(n) = O(n
2
), A is said
to be quadratic; and if f(n) ≥ O(e
n
), A is said to
be exponential. In general, if f(n) ≤ O(n
r
) for some
r ∈ R, then A is said to be polynomial.
As a standard terminology, mainly used in the
“Theory of Complexity”, polynomial algorithms are
also called tractable, while exponential algorithms are
called intractable. These names are due to the follow-
ing observation. Suppose we have a linear algorithm
A and an exponential algorithm B, not necessarily
solving the same problem. Also suppose that an hour
of computer time executes both A and B with n = 40.
If a new computer is used which is 1000 times faster
than the old one, in an hour algorithm A will exe-
cute with m such that f(m) = K
A
m = 1000K
A
n, or
m = 1000n = 40, 000. Therefore, the problem solved
by the new computer is 1000 times larger than the
problem solved by the old computer. For algorithm
B we have g(m) = K
B
e
m
= 1000K
B
e
n
; by simplify-
ing and taking logarithms, we find m = n+ln 1000 ≈
n + 6.9 < 47. Therefore, the improvement achieved
by the new computer is almost negligible.
Chapter 2
Special numbers
A sequence is a mapping from the set N of nat-
ural numbers into some other set of numbers. If
f : N → R, the sequence is called a sequence of real
numbers; if f : N → Q, the sequence is called a se-
quence of rational numbers; and so on. Usually, the
image of a k ∈ N is denoted by f
k
instead of the tradi-
tional f(k), and the whole sequence is abbreviated as
(f
0
, f
1
, f
2
, . . .) = (f
k
)
k∈N
. Because of this notation,
an element k ∈ N is called an index.
Often, we also study double sequences, i.e., map-
pings f : N N → R or some other nu-
meric set. In this case also, instead of writing
f(n, k) we will usually write f
n,k
and the whole
sequence will be denoted by ¦f
n,k
[ n, k ∈ N¦ or
(f
n,k
)
n,k∈N
. A double sequence can be displayed
as an infinite array of numbers, whose first row is
the sequence (f
0,0
, f
0,1
, f
0,2
, . . .), the second row is
(f
1,0
, f
1,1
, f
1,2
, . . .), and so on. The array can also
be read by columns, and then (f
0,0
, f
1,0
, f
2,0
, . . .) is
the first column, (f
0,1
, f
1,1
, f
2,1
, . . .) is the second col-
umn, and so on. Therefore, the index n is the row
index and the index k is the column index.
We wish to describe here some sequences and dou-
ble sequences of numbers, frequently occurring in the
analysis of algorithms. In fact, they arise in the study
of very simple and basic combinatorial problems, and
therefore appear in more complex situations, so to be
considered as the fundamental bricks in a solid wall.
2.1 Mappings and powers
In all branches of Mathematics two concepts appear,
which have to be taken as basic: the concept of a
set and the concept of a mapping. A set is a col-
lection of objects and it must be considered a primi-
tive concept, i.e., a concept which cannot be defined
in terms of other and more elementary concepts. If
a, b, c, . . . are the objects (or elements) in a set de-
noted by A, we write A = ¦a, b, c, . . .¦, thus giving
an extensional definition of this particular set. If a
set B is defined through a property P of its elements,
we write B = ¦x[ P(x) is true¦, thus giving an inten-
sional definition of the particular set B.
If S is a finite set, then [S[ denotes its cardinality
or the number of its elements: The order in which we
write or consider the elements of S is irrelevant. If we
wish to emphasize a particular arrangement or order-
ing of the elements in S, we write (a
1
, a
2
, . . . , a
n
), if
[S[ = n and S = ¦a
1
, a
2
, . . . , a
n
¦ in any order. This is
the vector notation and is used to represent arrange-
ments of S. Two arrangements of S are different if
and only if an index k exists, for which the elements
corresponding to that index in the two arrangements
are different; obviously, as sets, the two arrangements
continue to be the same.
If A, B are two sets, a mapping or function from A
into B, noted as f : A → B, is a subset of the Carte-
sian product of A by B, f ⊆ A B, such that every
element a ∈ A is the first element of one and only
one pair (a, b) ∈ f. The usual notation for (a, b) ∈ f
is f(a) = b, and b is called the image of a under the
mapping f. The set A is the domain of the map-
ping, while B is the range or codomain. A function
for which every a
1
= a
2
∈ A corresponds to pairs
(a
1
, b
1
) and (a
2
, b
2
) with b
1
= b
2
is called injective. A
function in which every b ∈ B belongs at least to one
pair (a, b) is called surjective. A bijection or 1-1 cor-
respondence or 1-1 mapping is any injective function
which is also surjective.
If [A[ = n and [B[ = m, the cartesian product
AB, i.e., the set of all the couples (a, b) with a ∈ A
and b ∈ B, contains exactly nm elements or couples.
A more difficult problem is to find out how many
mappings from A to B exist. We can observe that
every element a ∈ A must have its image in B; this
means that we can associate to a any one of the m
elements in B. Therefore, we have m m . . . m
different possibilities, when the product is extended
to all the n elements in A. Since all the mappings can
be built in this way, we have a total of m
n
different
mappings from A into B. This also explains why the
set of mappings from A into B is often denoted by
11
12 CHAPTER 2. SPECIAL NUMBERS
B
A
; in fact we can write:


B
A


= [B[
|A|
= m
n
.
This formula allows us to solve some simple com-
binatorial problems. For example, if we toss 5 coins,
how many different configurations head/tail are pos-
sible? The five coins are the domain of our mappings
and the set ¦head,tail¦ is the codomain. Therefore
we have a total of 2
5
= 32 different configurations.
Similarly, if we toss three dice, the total number of
configurations is 6
3
= 216. In the same way, we can
count the number of subsets in a set S having [S[ = n.
In fact, let us consider, given a subset A ⊆ S, the
mapping χ
A
: S → ¦0, 1¦ defined by:

χ
A
(x) = 1 for x ∈ A
χ
A
(x) = 0 for x / ∈ A
This is called the characteristic function of the sub-
set A; every two different subsets of S have dif-
ferent characteristic functions, and every mapping
f : S → ¦0, 1¦ is the characteristic function of some
subset A ⊆ S, i.e., the subset ¦x ∈ S [ f(x) = 1¦.
Therefore, there are as many subsets of S as there
are characteristic functions; but these are 2
n
by the
formula above.
A finite set A is sometimes called an alphabet and
its elements symbols or letters. Any sequence of let-
ters is called a word; the empty sequence is the empty
word and is denoted by ǫ. From the previous consid-
erations, if [A[ = n, the number of words of length m
is n
m
. The first part of Chapter 6 is devoted to some
basic notions on special sets of words or languages.
2.2 Permutations
In the usual sense, a permutation of a set of objects
is an arrangement of these objects in any order. For
example, three objects, denoted by a, b, c, can be ar-
ranged into six different ways:
(a, b, c), (a, c, b), (b, a, c), (b, c, a), (c, a, b), (c, b, a).
A very important problem in Computer Science is
“sorting”: suppose we have n objects from an or-
dered set, usually some set of numbers or some set of
strings (with the common lexicographic order); the
objects are given in a random order and the problem
consists in sorting them according to the given order.
For example, by sorting (60, 51, 80, 77, 44) we should
obtain (44, 51, 60, 77, 80) and the real problem is to
obtain this ordering in the shortest possible time. In
other words, we start with a random permutation of
the n objects, and wish to arrive to their standard
ordering, the one in accordance with their supposed
order relation (e.g., “less than”).
In order to abstract from the particular nature of
the n objects, we will use the numbers ¦1, 2, . . . , n¦ =
N
n
, and define a permutation as a 1-1 mapping π :
N
n
→ N
n
. By identifying a and 1, b and 2, c and 3,
the six permutations of 3 objects are written:

1 2 3
1 2 3

,

1 2 3
1 3 2

,

1 2 3
2 1 3

,

1 2 3
2 3 1

,

1 2 3
3 1 2

,

1 2 3
3 2 1

,
where, conventionally, the first line contains the el-
ements in N
n
in their proper order, and the sec-
ond line contains the corresponding images. This
is the usual representation for permutations, but
since the first line can be understood without
ambiguity, the vector representation for permuta-
tions is more common. This consists in writ-
ing the second line (the images) in the form of
a vector. Therefore, the six permutations are
(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), re-
spectively.
Let us examine the permutation π = (3, 2, 1) for
which we have π(1) = 3, π(2) = 2 and π(3) = 1.
If we start with the element 1 and successively ap-
ply the mapping π, we have π(1) = 3, π(π(1)) =
π(3) = 1, π(π(π(1))) = π(1) = 3 and so on. Since
the elements in N
n
are finite, by starting with any
k ∈ N
n
we must obtain a finite chain of numbers,
which will repeat always in the same order. These
numbers are said to form a cycle and the permuta-
tion (3, 2, 1) is formed by two cycles, the first one
composed by 1 and 3, the second one only composed
by 2. We write (3, 2, 1) = (1 3)(2), where every cycle
is written between parentheses and numbers are sep-
arated by blanks, to distinguish a cycle from a vector.
Conventionally, a cycle is written with the smallest
element first and the various cycles are arranged ac-
cording to their first element. Therefore, in this cycle
representation the six permutations are:
(1)(2)(3), (1)(2 3), (1 2)(3), (1 2 3), (1 3 2), (1 3)(2).
A number k for which π(k) = k is called a fixed
point for π. The corresponding cycles, formed by a
single element, are conventionally understood, except
in the identity (1, 2, . . . , n) = (1)(2) (n), in which
all the elements are fixed points; the identity is simply
written (1). Consequently, the usual representation
of the six permutations is:
(1) (2 3) (1 2) (1 2 3) (1 3 2) (1 3).
A permutation without any fixed point is called a
derangement. A cycle with only two elements is called
a transposition. The degree of a cycle is the number
of its elements, plus one; the degree of a permutation
2.3. THE GROUP STRUCTURE 13
is the sum of the degrees of its cycles. The six per-
mutations have degree 2, 3, 3, 4, 4, 3, respectively. A
permutation is even or odd according to the fact that
its degree is even or odd.
The permutation (8, 9, 4, 3, 6, 1, 7, 2, 10, 5),
in vector notation, has a cycle representation
(1 8 2 9 10 5 6)(3 4), the number 7 being a fixed
point. The long cycle (1 8 2 9 10 5 6) has degree 8;
therefore the permutation degree is 8 + 3 + 2 = 13
and the permutation is odd.
2.3 The group structure
Let n ∈ N; P
n
denotes the set of all the permutations
of n elements, i.e., according to the previous sections,
the set of 1-1 mappings π : N
n
→ N
n
. If π, ρ ∈ P
n
,
we can perform their composition, i.e., a new permu-
tation σ defined as σ(k) = π(ρ(k)) = (π ◦ ρ)(k). An
example in P
7
is:
π ◦ ρ =

1 2 3 4 5 6 7
1 5 6 7 4 2 3

1 2 3 4 5 6 7
4 5 2 1 7 6 3

=

1 2 3 4 5 6 7
4 7 6 3 1 5 2

.
In fact, by instance, π(2) = 5 and ρ(5) = 7; therefore
σ(2) = ρ(π(2)) = ρ(5) = 7, and so on. The vec-
tor representation of permutations is not particularly
suited for hand evaluation of composition, although it
is very convenient for computer implementation. The
opposite situation occurs for cycle representation:
(2 5 4 7 3 6) ◦ (1 4)(2 5 7 3) = (1 4 3 6 5)(2 7)
Cycles in the left hand member are read from left to
right and by examining a cycle after the other we find
the images of every element by simply remembering
the cycle successor of the same element. For example,
the image of 4 in the first cycle is 7; the second cycle
does not contain 7, but the third cycle tells us that the
image of 7 is 3. Therefore, the image of 4 is 3. Fixed
points are ignored, and this is in accordance with
their meaning. The composition symbol ‘◦’ is usually
understood and, in reality, the simple juxtaposition
of the cycles can well denote their composition.
The identity mapping acts as an identity for com-
position, since it is only composed by fixed points.
Every permutation has an inverse, which is the per-
mutation obtained by reading the given permutation
from below, i.e., by sorting the elements in the second
line, which become the elements in the first line, and
then rearranging the elements in the first line into
the second line. For example, the inverse of the first
permutation π in the example above is:

1 2 3 4 5 6 7
1 6 7 5 2 3 4

.
In fact, in cycle notation, we have:
(2 5 4 7 3 6)(2 6 3 7 4 5) =
= (2 6 3 7 4 5)(2 5 4 7 3 6) = (1).
A simple observation is that the inverse of a cycle
is obtained by writing its first element followed by
all the other elements in reverse order. Hence, the
inverse of a transposition is the same transposition.
Since composition is associative, we have proved
that (P
n
, ◦) is a group. The group is not commuta-
tive, because, for example:
ρ ◦ π = (1 4)(2 5 7 3) ◦ (2 5 4 7 3 6)
= (1 7 6 2 4)(3 5) = π ◦ ρ.
An involution is a permutation π such that π
2
= π ◦
π = (1). An involution can only be composed by fixed
points and transpositions, because by the definition
we have π
−1
= π and the above observation on the
inversion of cycles shows that a cycle with more than
2 elements has an inverse which cannot coincide with
the cycle itself.
Till now, we have supposed that in the cycle repre-
sentation every number is only considered once. How-
ever, if we think of a permutation as the product of
cycles, we can imagine that its representation is not
unique and that an element k ∈ N
n
can appear in
more than one cycle. The representation of σ or π ◦ρ
are examples of this statement. In particular, we can
obtain the transposition representation of a permuta-
tion; we observe that we have:
(2 6)(6 5)(6 4)(6 7)(6 3) = (2 5 4 7 3 6)
We transform the cycle into a product of transposi-
tions by forming a transposition with the first and
the last element in the cycle, and then adding other
transpositions with the same first element (the last
element in the cycle) and the other elements in the
cycle, in the same order, as second element. Besides,
we note that we can always add a couple of transpo-
sitions as (2 5)(2 5), corresponding to the two fixed
points (2) and (5), and therefore adding nothing to
the permutation. All these remarks show that:
• every permutation can be written as the compo-
sition of transpositions;
• this representation is not unique, but every
two representations differ by an even number of
transpositions;
• the minimal number of transpositions corre-
sponding to a cycle is the degree of the cycle
(except possibly for fixed points, which however
always correspond to an even number of trans-
positions).
14 CHAPTER 2. SPECIAL NUMBERS
Therefore, we conclude that an even [odd] permuta-
tion can be expressed as the composition of an even
[odd] number of transpositions. Since the composi-
tion of two even permutations is still an even permu-
tation, the set A
n
of even permutations is a subgroup
of P
n
and is called the alternating subgroup, while the
whole group P
n
is referred to as the symmetric group.
2.4 Counting permutations
How many permutations are there in P
n
? If n = 1,
we only have a single permutation (1), and if n = 2
we have two permutations, exactly (1, 2) and (2, 1).
We have already seen that [P
3
[ = 6 and if n = 0
we consider the empty vector () as the only possible
permutation, that is [P
0
[ = 1. In this way we obtain
a sequence ¦1, 1, 2, 6, . . .¦ and we wish to obtain a
formula giving us [P
n
[, for every n ∈ N.
Let π ∈ P
n
be a permutation and (a
1
, a
2
, ..., a
n
) be
its vector representation. We can obtain a permuta-
tion in P
n+1
by simply adding the new element n+1
in any position of the representation of π:
(n + 1, a
1
, a
2
, . . . , a
n
) (a
1
, n + 1, a
2
, . . . , a
n
)
(a
1
, a
2
, . . . , a
n
, n + 1)
Therefore, from any permutation in P
n
we obtain
n+1 permutations in P
n+1
, and they are all different.
Vice versa, if we start with a permutation in P
n+1
,
and eliminate the element n + 1, we obtain one and
only one permutation in P
n
. Therefore, all permuta-
tions in P
n+1
are obtained in the way just described
and are obtained only once. So we find:
[P
n+1
[ = (n + 1) [P
n
[
which is a simple recurrence relation. By unfolding
this recurrence, i.e., by substituting to [P
n
[ the same
expression in [P
n+1
[, and so on, we obtain:
[P
n+1
[ = (n + 1)[P
n
[ = (n + 1)n[P
n−1
[ =
= =
= (n + 1)n(n −1) 1 [P
0
[
Since, as we have seen, [P
0
[=1, we have proved that
the number of permutations in P
n
is given by the
product n (n − 1)... 2 1. Therefore, our sequence
is:
n 0 1 2 3 4 5 6 7 8
[P
n
[ 1 1 2 6 24 120 720 5040 40320
As we mentioned in the Introduction, the number
n (n−1)... 2 1 is called n factorial and is denoted by
n!. For example we have 10! = 10987654321 =
3, 680, 800. Factorials grow very fast, but they are one
of the most important quantities in Mathematics.
When n ≥ 2, we can add to every permutation in
P
n
one transposition, say (1 2). This transforms ev-
ery even permutation into an odd permutation, and
vice versa. On the other hand, since (1 2)
−1
= (1 2),
the transformation is its own inverse, and therefore
defines a 1-1 mapping between even and odd permu-
tations. This proves that the number of even (odd)
permutations is n!/2.
Another simple problem is how to determine the
number of involutions on n elements. As we have al-
ready seen, an involution is only composed by fixed
points and transpositions (without repetitions of the
elements!). If we denote by I
n
the set of involutions
of n elements, we can divide I
n
into two subsets: I
′
n
is the set of involutions in which n is a fixed point,
and I
′′
n
is the set of involutions in which n belongs
to a transposition, say (k n). If we eliminate n from
the involutions in I
′
n
, we obtain an involution of n−1
elements, and vice versa every involution in I
′
n
can be
obtained by adding the fixed point n to an involution
in I
n−1
. If we eliminate the transposition (k n) from
an involution in I
′′
n
, we obtain an involution in I
n−2
,
which contains the element n−1, but does not contain
the element k. In all cases, however, by eliminating
(k n) from all involutions containing it, we obtain a
set of involutions in a 1-1 correspondence with I
n−2
.
The element k can assume any value 1, 2, . . . , n − 1,
and therefore we obtain (n − 1) times [I
n−2
[ involu-
tions.
We now observe that all the involutions in I
n
are
obtained in this way from involutions in I
n−1
and
I
n−2
, and therefore we have:
[I
n
[ = [I
n−1
[ + (n −1)[I
n−2
[
Since [I
0
[ = 1, [I
1
[ = 1 and [I
2
[ = 2, from this recur-
rence relation we can successively find all the values
of [I
n
[. This sequence (see Section 4.9) is therefore:
n 0 1 2 3 4 5 6 7 8
I
n
1 1 2 4 10 26 76 232 764
We conclude this section by giving the classical
computer program for generating a random permu-
tation of the numbers ¦1, 2, . . . , n¦. The procedure
shuffle receives the address of the vector, and the
number of its elements; fills it with the numbers from
1 to n then uses the standard procedure random to
produce a random permutation, which is returned in
the input vector:
procedure shuffle( var v : vector; n : integer ) ;
var : ;
begin
for i := 1 to n do v[ i ] := i ;
for i := n downto 2 do begin
j := random( i ) + 1 ;
2.5. DISPOSITIONS AND COMBINATIONS 15
a := v[ i ]; v[ i ] := v[ j ]; v[ j ] := a
end
end ¦shuffle¦ ;
The procedure exchanges the last element in v with
a random element in v, possibly the same last ele-
ment. In this way the last element is chosen at ran-
dom, and the procedure goes on with the last but one
element. In this way, the elements in v are properly
shuffled and eventually v contains the desired random
permutation. The procedure obviously performs in
linear time.
2.5 Dispositions and Combina-
tions
Permutations are a special case of a more general sit-
uation. If we have n objects, we can wonder how
many different orderings exist of k among the n ob-
jects. For example, if we have 4 objects a, b, c, d, we
can make 12 different arrangements with two objects
chosen from a, b, c, d. They are:
(a, b) (a, c) (a, d) (b, a) (b, c) (b, d)
(c, a) (c, b) (c, d) (d, a) (d, b) (d, c)
These arrangements ara called dispositions and, in
general, we can use any one of the n objects to be first
in the permutation. There remain only n −1 objects
to be used as second element, and n − 2 objects to
be used as a third element, and so on. Therefore,
the k objects can be selected in n(n−1)...(n−k +1)
different ways. If D
n,k
denotes the number of possible
dispositions of n elements in groups of k, we have:
D
n,k
= n(n −1) (n −k + 1) = n
k
The symbol n
k
is called a falling factorial because it
consists of k factors beginning with n and decreasing
by one down to (n −k +1). Obviously, n
n
= n! and,
by convention, n
0
= 1. There exists also a rising
factorial n
¯
k
= n(n + 1) (n +k −1), often denoted
by (n)
k
, the so-called Pochhammer symbol.
When in k-dispositions we do not consider the
order of the elements, we obtain what are called
k-combinations. Therefore, there are only 6 2-
combinations of 4 objects, and they are:
¦a, b¦ ¦a, c¦ ¦a, d¦ ¦b, c¦ ¦b, d¦ ¦c, d¦
Combinations are written between braces, because
they are simply the subsets with k objects of a set of n
objects. If A = ¦a, b, c¦, all the possible combinations
of these objects are:
C
3,0
= ∅
C
3,1
= ¦¦a¦, ¦b¦, ¦c¦¦
C
3,2
= ¦¦a, b¦, ¦a, c¦, ¦b, c¦¦
C
3,3
= ¦¦a, b, c¦¦
The number of k-combinations of n objects is denoted
by

n
k

, which is often read “n choose k”, and is called
a binomial coefficient. By the very definition we have:

n
0

= 1

n
n

= 1 ∀n ∈ N
because, given a set of n elements, the empty set is
the only subset with 0 elements, and the whole set is
the only subset with n elements.
The name “binomial coefficients” is due to the well-
known “binomial formula”:
(a +b)
n
=
n
¸
k=0

n
k

a
k
b
n−k
which is easily proved. In fact, when expanding the
product (a + b)(a + b) (a + b), we choose a term
from each factor (a +b); the resulting term a
k
b
n−k
is
obtained by summing over all the possible choices of
k a’s and n − k b’s, which, by the definition above,
are just

n
k

.
There exists a simple formula to compute binomial
coefficients. As we have seen, there are n
k
differ-
ent k-dispositions of n objects; by permuting the k
objects in the disposition, we obtain k! other dispo-
sitions with the same elements. Therefore, k! dis-
positions correspond to a single combination and we
have:

n
k

=
n
k
k!
=
n(n −1) . . . (n −k + 1)
k!
This formula gives a simple way to compute binomial
coefficients in a recursive way. In fact we have:

n
k

=
n(n −1) . . . (n −k + 1)
k!
=
=
n
k
(n −1) . . . (n −k + 1)
(k −1)!
=
=
n
k

n −1
k −1

and the expression on the right is successively reduced
to

r
0

, which is 1 as we have already seen. For exam-
ple, we have:

7
3

=
7
3

6
2

=
7
3
6
2

5
1

=
7
3
6
2
5
1

4
0

= 35
It is not difficult to compute a binomial coefficient
such as

100
3

, but it is a hard job to compute

100
97

16 CHAPTER 2. SPECIAL NUMBERS
by means of the above formulas. There exists, how-
ever, a symmetry formula which is very useful. Let
us begin by observing that:

n
k

=
n(n −1) . . . (n −k + 1)
k!
=
=
n. . . (n −k + 1)
k!
(n −k) . . . 1
(n −k) . . . 1
=
=
n!
k!(n −k)!
This is a very important formula by its own, and it
shows that:

n
n −k

=
n!
(n −k)!(n −(n −k))!
=

n
k

From this formula we immediately obtain

100
97

=

100
3

. The most difficult computing problem is the
evaluation of the central binomial coefficient

2k
k

, for
which symmetry gives no help.
The reader is invited to produce a computer pro-
gram to evaluate binomial coefficients. He (or she) is
warned not to use the formula n!/k!(n−k)!, which can
produce very large numbers, exceeding the capacity
of the computer when n, k are not small.
The definition of a binomial coefficient can be eas-
ily expanded to any real numerator:

r
k

=
r
k
k!
=
r(r −1) . . . (r −k + 1)
k!
.
For example we have:

1/2
3

=
1/2(−1/2)(−3/2)
3!
=
1
16
but in this case the symmetry rule does not make
sense. We point out that:

−n
k

=
−n(−n −1) . . . (−n −k + 1)
k!
=
=
(−1)
k
(n +k −1)
k
k!
=

n +k −1
k

(−1)
k
which allows us to express a binomial coefficient with
negative, integer numerator as a binomial coefficient
with positive numerator. This is known as negation
rule and will be used very often.
If in a combination we are allowed to have several
copies of the same element, we obtain a combination
with repetitions. A useful exercise is to prove that the
number of the k by k combinations with repetitions
of n elements is:
R
n,k
=

n +k −1
k

.
2.6 The Pascal triangle
Binomial coefficients satisfy a very important recur-
rence relation, which we are now going to prove. As
we know,

n
k

is the number of the subsets with k
elements of a set with n elements. We can count
the number of such subsets in the following way. Let
¦a
1
, a
2
, . . . , a
n
¦ be the elements of the base set, and
let us fix one of these elements, e.g., a
n
. We can dis-
tinguish the subsets with k elements into two classes:
the subsets containing a
n
and the subsets that do
not contain a
n
. Let S
+
and S
−
be these two classes.
We now point out that S
+
can be seen (by elimi-
nating a
n
) as the subsets with k − 1 elements of a
set with n−1 elements; therefore, the number of the
elements in S
+
is

n−1
k−1

. The class S
−
can be seen
as composed by the subsets with k elements of a set
with n−1 elements, i.e., the base set minus a
n
: their
number is therefore

n−1
k

. By summing these two
contributions, we obtain the recurrence relation:

n
k

=

n −1
k

+

n −1
k −1

which can be used with the initial conditions:

n
0

=

n
n

= 1 ∀n ∈ N.
For example, we have:

4
2

=

3
2

+

3
1

=
=

2
2

+

2
1

+

2
1

+

2
0

=
= 2 + 2

2
1

= 2 + 2

1
1

+

1
0

=
= 2 + 2 2 = 6.
This recurrence is not particularly suitable for nu-
merical computation. However, it gives a simple rule
to compute successively all the binomial coefficients.
Let us dispose them in an infinite array, whose rows
represent the number n and whose columns represent
the number k. The recurrence tells us that the ele-
ment in position (n, k) is obtained by summing two
elements in the previous row: the element just above
the position (n, k), i.e., in position (n−1, k), and the
element on the left, i.e., in position (n−1, k−1). The
array is initially filled by 1’s in the first column (cor-
responding to the various

n
0

) and the main diagonal
(corresponding to the numbers

n
n

). See Table 2.1.
We actually obtain an infinite, lower triangular ar-
ray known as the Pascal triangle (or Tartaglia-Pascal
triangle). The symmetry rule is quite evident and a
simple observation is that the sum of the elements in
row n is 2
n
. The proof of this fact is immediate, since
2.7. HARMONIC NUMBERS 17
n`k 0 1 2 3 4 5 6
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
Table 2.1: The Pascal triangle
a row represents the total number of the subsets in a
set with n elements, and this number is obviously 2
n
.
The reader can try to prove that the row alternating
sums, e.g., the sum 1−4+6−4+1, equal zero, except
for the first row, the row with n = 0.
As we have already mentioned, the numbers c
n
=

2n
n

are called central binomial coefficients and their
sequence begins:
n 0 1 2 3 4 5 6 7 8
c
n
1 2 6 20 70 252 924 3432 12870
They are very important, and, for example, we can
express the binomial coefficients

−1/2
n

in terms of
the central binomial coefficients:

−1/2
n

=
(−1/2)(−3/2) (−(2n −1)/2)
n!
=
=
(−1)
n
1 3 (2n −1)
2
n
n!
=
=
(−1)
n
1 2 3 4 (2n −1) (2n)
2
n
n!2 4 (2n)
=
=
(−1)
n
(2n)!
2
n
n!2
n
(1 2 n)
=
(−1)
n
4
n
(2n)!
n!
2
=
=
(−1)
n
4
n

2n
n

.
In a similar way, we can prove the following identities:

1/2
n

=
(−1)
n−1
4
n
(2n −1)

2n
n


3/2
n

=
(−1)
n
3
4
n
(2n −1)(2n −3)

2n
n


−3/2
n

=
(−1)
n−1
(2n + 1)
4
n

2n
n

.
An important point of this generalization is that
the binomial formula can be extended to real expo-
nents. Let us consider the function f(x) = (1+x)
r
; it
is continuous and can be differentiated as many times
as we wish; in fact we have:
f
(n)
(x) =
d
n
dx
n
(1 +x)
r
= r
n
(1 +x)
r−n
as can be shown by mathematical induction. The
first two cases are f
(0)
= (1 + x)
r
= r
0
(1 + x)
r−0
,
f
′
(x) = r(1 + x)
r−1
. Suppose now that the formula
holds true for some n ∈ N and let us differentiate it
once more:
f
(n+1)
(x) = r
n
(r −n)(1 +x)
r−n−1
=
= r
n+1
(1 +x)
r−(n+1)
and this proves our statement. Because of that, (1 +
x)
r
has a Taylor expansion around the point x = 0
of the form:
(1 +x)
r
=
= f(0) +
f
′
(0)
1!
x +
f
′′
(0)
2!
x
2
+ +
f
(n)
(0)
n!
x
n
+ .
The coefficient of x
n
is therefore f
(n)
(0)/n! =
r
n
/n! =

r
n

, and so:
(1 +x)
r
=
∞
¸
n=0

r
n

x
n
.
We conclude with the following property, which is
called the cross-product rule:

n
k

k
r

=
n!
k!(n −k)!
k!
r!(k −r)!
=
=
n!
r!(n −r)!
(n −r)!
(n −k)!(k −r)!
=
=

n
r

n −r
k −r

.
This rule, together with the symmetry and the nega-
tion rules, are the three basic properties of binomial
coefficients:

n
k

=

n
n −k

−n
k

=

n +k −1
k

(−1)
k

n
k

k
r

=

n
r

n −r
k −r

and are to be remembered by heart.
2.7 Harmonic numbers
It is well-known that the harmonic series:
∞
¸
k=1
1
k
= 1 +
1
2
+
1
3
+
1
4
+
1
5
+
diverges. In fact, if we cumulate the 2
m
numbers
from 1/(2
m
+ 1) to 1/2
m+1
, we obtain:
1
2
m
+ 1
+
1
2
m
+ 2
+ +
1
2
m+1
>
>
1
2
m+1
+
1
2
m+1
+ +
1
2
m+1
=
2
m
2
m+1
=
1
2
18 CHAPTER 2. SPECIAL NUMBERS
and therefore the sum cannot be limited. On the
other hand we can define:
H
n
= 1 +
1
2
+
1
3
+ +
1
n
a finite, partial sum of the harmonic series. This
number has a well-defined value and is called a har-
monic number. Conventionally, we set H
0
= 0 and
the sequence of harmonic numbers begins:
n 0 1 2 3 4 5 6 7 8
H
n
0 1
3
2
11
6
25
12
137
60
49
20
363
140
761
280
Harmonic numbers arise in the analysis of many
algorithms and it is very useful to know an approxi-
mate value for them. Let us consider the series:
1 −
1
2
+
1
3
−
1
4
+
1
5
− = ln 2
and let us define:
L
n
= 1 −
1
2
+
1
3
−
1
4
+ +
(−1)
n−1
n
.
Obviously we have:
H
2n
−L
2n
= 2

1
2
+
1
4
+ +
1
2n

= H
n
or H
2n
= L
2n
+H
n
, and since the series for ln 2 is al-
ternating in sign, the error committed by truncating
it at any place is less than the first discarded element.
Therefore:
ln 2 −
1
2n
< L
2n
< ln 2
and by summing H
n
to all members:
H
n
+ ln 2 −
1
2n
< H
2n
< H
n
+ ln 2
Let us now consider the two cases n = 2
k−1
and n =
2
k−2
:
H
2
k−1 + ln 2 −
1
2
k
< H
2
k < H
2
k−1 + ln 2
H
2
k−2 + ln 2 −
1
2
k−1
< H
2
k−1 < H
2
k−2 + ln 2.
By summing and simplifying these two expressions,
we obtain:
H
2
k−2 + 2 ln 2 −
1
2
k
−
1
2
k−1
< H
2
k < H
2
k−2 + 2 ln 2.
We can now iterate this procedure and eventually
find:
H
2
0 +k ln 2−
1
2
k
−
1
2
k−1
− −
1
2
1
< H
2
k < H
2
0 +k ln 2.
Since H
2
0 = H
1
= 1, we have the limitations:
ln 2
k
< H
2
k < ln 2
k
+ 1.
These limitations can be extended to every n, and
since the values of the H
n
’s are increasing, this im-
plies that a constant γ should exist (0 < γ < 1) such
that:
H
n
→ ln n +γ as n → ∞
This constant is called the Euler-Mascheroni con-
stant and, as we have already mentioned, its value
is: γ ≈ 0.5772156649 . . .. Later we will prove the
more accurate approximation of the H
n
’s we quoted
in the Introduction.
The generalized harmonic numbers are defined as:
H
(s)
n
=
1
1
s
+
1
2
s
+
1
3
s
+ +
1
n
s
and H
(1)
n
= H
n
. They are the partial sums of the
series defining the Riemann ζ function:
ζ(s) =
1
1
s
+
1
2
s
+
1
3
s
+
which can be defined in such a way that the sum
actually converges except for s = 1 (the harmonic
series). In particular we have:
ζ(2) = 1 +
1
4
+
1
9
+
1
16
+
1
25
+ =
π
2
6
ζ(3) = 1 +
1
8
+
1
27
+
1
64
+
1
125
+
ζ(4) = 1 +
1
16
+
1
81
+
1
256
+
1
625
+ =
π
4
90
and in general:
ζ(2n) =
(2π)
2n
2(2n)!
[B
2n
[
where B
n
are the Bernoulli numbers (see below). No
explicit formula is known for ζ(2n + 1), but numeri-
cally we have:
ζ(3) ≈ 1.202056903 . . . .
Because of the limited value of ζ(s), we can set, for
large values of n:
H
(s)
n
≈ ζ(s)
2.8 Fibonacci numbers
At the beginning of 1200, Leonardo Fibonacci intro-
duced in Europe the positional notation for numbers,
together with the computing algorithms for perform-
ing the four basic operations. In fact, Fibonacci was
2.9. WALKS, TREES AND CATALAN NUMBERS 19
the most important mathematician in western Eu-
rope at that time. He posed the following problem:
a farmer has a couple of rabbits which generates an-
other couple after two months and, from that moment
on, a new couple of rabbits every month. The new
generated couples become fertile after two months,
when they begin to generate a new couple of rab-
bits every month. The problem consists in comput-
ing how many couples of rabbits the farmer has after
n months.
It is a simple matter to find the initial values: there
is one couple at the beginning (1st month) and 1 in
the second month. The third month the farmer has
2 couples, and 3 couples the fourth month; in fact,
the first couple has generated another pair of rabbits,
while the previously generated couple of rabbits has
not yet become fertile. The couples become 5 on the
fifth month: in fact, there are the 3 couples of the
preceding month plus the newly generated couples,
which are as many as there are fertile couples; but
these are just the couples of two months beforehand,
i.e., 2 couples. In general, at the nth month, the
farmer will have the couples of the previous month
plus the new couples, which are generated by the fer-
tile couples, that is the couples he had two months
before. If we denote by F
n
the number of couples at
the nth month, we have the Fibonacci recurrence:
F
n
= F
n−1
+F
n−2
with the initial conditions F
1
= F
2
= 1. By the same
rule, we have F
0
= 0 and the sequence of Fibonacci
numbers begins:
n 0 1 2 3 4 5 6 7 8 9 10
F
n
0 1 1 2 3 5 8 13 21 34 55
every term obtained by summing the two preceding
numbers in the sequence.
Despite the small numbers appearing at the begin-
ning of the sequence, Fibonacci numbers grow very
fast, and later we will see how they grow and how they
can be computed in a fast way. For the moment, we
wish to show how Fibonacci numbers appear in com-
binatorics and in the analysis of algorithms. Suppose
we have some bricks of dimensions 1 2 dm, and we
wish to cover a strip 2 dm wide and n dm long by
using these bricks. The problem is to know in how
many different ways we can perform this covering. In
Figure 2.1 we show the five ways to cover a strip 4
dm long.
If M
n
is this number, we can observe that a cov-
ering of M
n
can be obtained by adding vertically a
brick to a covering in M
n−1
or by adding horizon-
tally two bricks to a covering in M
n−2
. These are the
only ways of proceeding to build our coverings, and
Figure 2.1: Fibonacci coverings for a strip 4 dm long
therefore we have the recurrence relation:
M
n
= M
n−1
+M
n−2
which is the same recurrence as for Fibonacci num-
bers. This time, however, we have the initial condi-
tions M
0
= 1 (the empty covering is just a covering!)
and M
1
= 1. Therefore we conclude M
n
= F
n+1
.
Euclid’s algorithm for computing the Greatest
Common Divisor (gcd) of two positive integer num-
bers is another instance of the appearance of Fi-
bonacci numbers. The problem is to determine the
maximal number of divisions performed by Euclid’s
algorithm. Obviously, this maximum is attained
when every division in the process gives 1 as a quo-
tient, since a greater quotient would drastically cut
the number of necessary divisions. Let us consider
two consecutive Fibonacci numbers, for example 34
and 55, and let us try to find gcd(34, 55):
55 = 1 34 + 21
34 = 1 21 + 13
21 = 1 13 + 8
13 = 1 8 + 5
8 = 1 5 + 3
5 = 1 3 + 2
3 = 1 2 + 1
2 = 2 1
We immediately see that the quotients are all 1 (ex-
cept the last one) and the remainders are decreasing
Fibonacci numbers. The process can be inverted to
prove that only consecutive Fibonacci numbers enjoy
this property. Therefore, we conclude that given two
integer numbers, n and m, the maximal number of
divisions performed by Euclid’s algorithm is attained
when n, m are two consecutive Fibonacci numbers
and the actual number of divisions is the order of the
smaller number in the Fibonacci sequence, minus 1.
2.9 Walks, trees and Catalan
numbers
“Walks” or “paths” are common combinatorial ob-
jects and are defined in the following way. Let Z
2
be
20 CHAPTER 2. SPECIAL NUMBERS
T
E




















, ,
, ,
,
, , , ,
, , ,
,
,
,
,
A
B
C
D
Figure 2.2: How a walk is decomposed
the integral lattice, i.e., the set of points in R
2
having
integer coordinates. A walk or path is a finite sequence
of points in Z
2
with the following properties:
1. the origin (0, 0) belongs to the walk;
2. if (x + 1, y + 1) belongs to the walk, then either
(x, y + 1) or (x + 1, y) also belongs to the walk.
A pair of points ((x, y), (x + 1, y)) is called an east
step and a pair ((x, y), (x, y + 1)) is a north step. It
is a simple matter to show that the number of walks
composed by n steps and ending at column k (i.e.,
the last point is (x, k) = (n − k, k)) is just

n
k

. In
fact, if we denote by 1, 2, . . . , n the n steps, starting
with the origin, then we can associate to any walk a
subset of N
n
= ¦1, 2, . . . , n¦, that is the subset of the
north-step numbers. Since the other steps should be
east steps, a 1-1 correspondence exists between the
walks and the subsets of N
n
with k elements, which,
as we know, are exactly

n
k

. This also proves, in a
combinatorial way, the symmetry rule for binomial
coefficients.
Among the walks, the ones that never go above the
main diagonal, i.e., walks no point of which has the
form (x, y) with y > x, are particularly important.
They are called underdiagonal walks. Later on, we
will be able to count the number of underdiagonal
walks composed by n steps and ending at a hori-
zontal distance k from the main diagonal. For the
moment, we only wish to establish a recurrence rela-
tion for the number b
n
of the underdiagonal walks of
length 2n and ending on the main diagonal. In Fig.
2.2 we sketch a possible walk, where we have marked
the point C = (k, k) at which the walk encounters
the main diagonal for the first time. We observe that
the walk is decomposed into a first part (the walk be-
tween A and B) and a second part (the walk between
C and D) which are underdiagonal walks of the same
type. There are b
k−1
walks of the type AB and b
n−k
walks of the type CD and, obviously, k can be any
number between 1 and n. We therefore obtain the
recurrence relation:
b
n
=
n
¸
k=1
b
k−1
b
n−k
=
n−1
¸
k=0
b
k
b
n−k−1
with the initial condition b
0
= 1, corresponding to
the empty walk or the walk composed by the only
point (0, 0). The sequence (b
k
)
k∈N
begins:
n 0 1 2 3 4 5 6 7 8
b
n
1 1 2 5 14 42 132 429 1430
and, as we shall see:
b
n
=
1
n + 1

2n
n

.
The b
n
’s are called Catalan numbers and they fre-
quently occur in the analysis of algorithms and data
structures. For example, if we associate an open
parenthesis to every east steps and a closed paren-
thesis to every north step, we obtain the number of
possible parenthetizations of an expression. When
we have three pairs of parentheses, the 5 possibilities
are:
()()() ()(()) (())() (()()) ((())).
When we build binary trees from permutations, we
do not always obtain different trees from different
permutations. There are only 5 trees generated by
the six permutations of 1, 2, 3, as we show in Figure
2.3.
How many different trees exist with n nodes? If we
fix our attention on the root, the left subtree has k
nodes for k = 0, 1, . . . , n − 1, while the right subtree
contains the remaining n − k − 1 nodes. Every tree
with k nodes can be combined with every tree with
n − k − 1 nodes to form a tree with n nodes, and
therefore we have the recurrence relation:
b
n
=
n−1
¸
k=0
b
k
b
n−k−1
which is the same recurrence as before. Since the
initial condition is again b
0
= 1 (the empty tree) there
are as many trees as there are walks.
Another kind of walks is obtained by considering
steps of type ((x, y), (x + 1, y + 1)), i.e., north-east
steps, and of type ((x, y), (x + 1, y − 1)), i.e., south-
east steps. The interesting walks are those starting
2.10. STIRLING NUMBERS OF THE FIRST KIND 21
1
2
3
d
d
d
d
1
3
2
d
d


1
2
3

d
d
1
3
2


d
d
1
2
3




Figure 2.3: The five binary trees with three nodes









d
d








d
d



d
d



d
d



d
d
Figure 2.4: Rooted Plane Trees
from the origin and never going below the x-axis; they
are called Dyck walks. An obvious 1-1 correspondence
exists between Dyck walks and the walks considered
above, and again we obtain the sequence of Catalan
numbers:
n 0 1 2 3 4 5 6 7 8
b
n
1 1 2 5 14 42 132 429 1430
Finally, the concept of a rooted planar tree is as
follows: let us consider a node, which is the root of
the tree; if we recursively add branches to the root or
to the nodes generated by previous insertions, what
we obtain is a “rooted plane tree”. If n denotes the
number of branches in a rooted plane tree, in Fig. 2.4
we represent all the trees up to n = 3. Again, rooted
plane trees are counted by Catalan numbers.
2.10 Stirling numbers of the
first kind
About 1730, the English mathematician James Stir-
ling was looking for a connection between powers
of a number x, say x
n
, and the falling factorials
x
k
= x(x − 1) (x − k + 1). He developed the first
n`k 0 1 2 3 4 5 6
0 1
1 0 1
2 0 1 1
3 0 2 3 1
4 0 6 11 6 1
5 0 24 50 35 10 1
6 0 120 274 225 85 15 1
Table 2.2: Stirling numbers of the first kind
instances:
x
1
= x
x
2
= x(x −1) = x
2
−x
x
3
= x(x −1)(x −2) = x
3
−3x
2
+ 2x
x
4
= x(x −1)(x −2)(x −3) =
= x
4
−6x
3
+ 11x
2
−6x
and picking the coefficients in their proper order
(from the smallest power to the largest) he obtained a
table of integer numbers. We are mostly interested in
the absolute values of these numbers, as are shown in
Table 2.2. After him, these numbers are called Stir-
ling numbers of the first kind and are now denoted by

n
k

, sometimes read “n cycle k”, for the reason we
are now going to explain.
First note that the above identities can be written:
x
n
=
n
¸
k=0

n
k

(−1)
n−k
x
k
.
Let us now observe that x
n
= x
n−1
(x − n + 1) and
therefore we have:
x
n
= (x −n + 1)
n−1
¸
k=0
¸
n −1
k

(−1)
n−k−1
x
k
=
=
n−1
¸
k=0
¸
n −1
k

(−1)
n−k−1
x
k+1
−
−
n−1
¸
k=0
(n −1)
¸
n −1
k

(−1)
n−k−1
x
k
=
=
n
¸
k=0
¸
n −1
k −1

(−1)
n−k
x
k
+
22 CHAPTER 2. SPECIAL NUMBERS
+
n
¸
k=0
(n −1)
¸
n −1
k

(−1)
n−k
x
k
.
We performed the change of variable k → k − 1 in
the first sum and then extended both sums from 0
to n. This identity is valid for every value of x, and
therefore we can equate its coefficients to those of the
previous, general Stirling identity, thus obtaining the
recurrence relation:

n
k

= (n −1)
¸
n −1
k

+
¸
n −1
k −1

.
This recurrence, together with the initial conditions:

n
n

= 1, ∀n ∈ N and

n
0

= 0, ∀n > 0,
completely defines the Stirling number of the first
kind.
What is a possible combinatorial interpretation of
these numbers? Let us consider the permutations of
n elements and count the permutations having ex-
actly k cycles, whose set will be denoted by S
n,k
. If
we fix any element, say the last element n, we ob-
serve that the permutations in S
n,k
can have n as a
fixed point, or not. When n is a fixed point and we
eliminate it, we obtain a permutation with n − 1 el-
ements having exactly k − 1 cycles; vice versa, any
such permutation gives a permutation in S
n,k
with n
as a fixed point if we add (n) to it. Therefore, there
are [S
n−1,k−1
[ such permutations in S
n,k
. When n is
not a fixed point and we eliminate it from the permu-
tation, we obtain a permutation with n −1 elements
and k cycles. However, the same permutation is ob-
tained several times, exactly n−1 times, since n can
occur after any other element in the standard cycle
representation (it can never occur as the first element
in a cycle, by our conventions). For example, all the
following permutations in S
5,2
produce the same per-
mutation in S
4,2
:
(1 2 3)(4 5) (1 2 3 5)(4) (1 2 5 3)(4) (1 5 2 3)(4).
The process can be inverted and therefore we have:
[S
n,k
[ = (n −1)[S
n−1,k
[ +[S
n−1,k−1
[
which is just the recurrence relation for the Stirling
numbers of the first kind. If we now prove that also
the initial conditions are the same, we conclude that
[S
n,k
[ =

n
k

. First we observe that S
n,n
is only com-
posed by the identity, the only permutation having n
cycles, i.e., n fixed points; so [S
n,n
[ = 1, ∀n ∈ N.
Moreover, for n ≥ 1, S
n,0
is empty, because ev-
ery permutation contains at least one cycle, and so
[S
n,0
[ = 0. This concludes the proof.
As an immediate consequence of this reasoning, we
find:
n
¸
k=0

n
k

= n!,
i.e., the row sums of the Stirling triangle of the first
kind equal n!, because they correspond to the total
number of permutations of n objects. We also observe
that:
•

n
1

= (n − 1)!; in fact, S
n,1
is composed by
all the permutations having a single cycle; this
begins by 1 and is followed by any permutations
of the n −1 remaining numbers;
•

n
n−1

=

n
2

; in fact, S
n,n−1
contains permu-
tations having all fixed points except a single
transposition; but this transposition can only be
formed by taking two elements among 1, 2, . . . , n,
which is done in

n
2

different ways;
•

n
2

= (n − 1)!H
n−1
; returning to the numeri-
cal definition, the coefficient of x
2
is a sum of
products, in each of which a positive integer is
missing.
2.11 Stirling numbers of the
second kind
James Stirling also tried to invert the process de-
scribed in the previous section, that is he was also
interested in expressing ordinary powers in terms of
falling factorials. The first instances are:
x
1
= x
1
x
2
= x
1
+x
2
= x +x(x −1)
x
3
= x
1
+ 3x
2
+x
3
= x + 3x(x −1)+
+x(x −1)(x −2)
x
4
= x
1
+ 6x
2
+ 7x
3
+x
4
The coefficients can be arranged into a triangular ar-
ray, as shown in Table 2.3, and are called Stirling
numbers of the second kind. The usual notation for
them is
¸
n
k
¸
, often read “n subset k”, for the reason
we are going to explain.
Stirling’s identities can be globally written:
x
n
=
n
¸
k=0

n
k
¸
x
k
.
We obtain a recurrence relation in the following way:
x
n
= xx
n−1
= x
n−1
¸
k=0

n −1
k

x
k
=
=
n−1
¸
k=0

n −1
k

(x +k −k)x
k
=
2.12. BELL AND BERNOULLI NUMBERS 23
n`k 0 1 2 3 4 5 6
0 1
1 0 1
2 0 1 1
3 0 1 3 1
4 0 1 7 6 1
5 0 1 15 25 10 1
6 0 1 31 90 65 15 1
Table 2.3: Stirling numbers of the second kind
=
n−1
¸
k=0

n −1
k

x
k+1
+
n−1
¸
k=0
k

n −1
k

x
k
=
=
n
¸
k=0

n −1
k −1

x
k
+
n
¸
k=0
k

n −1
k

x
k
where, as usual, we performed the change of variable
k → k − 1 and extended the two sums from 0 to n.
The identity is valid for every x ∈ R, and therefore we
can equate the coefficients of x
k
in this and the above
identity, thus obtaining the recurrence relation:

n
k
¸
= k

n −1
k

+

n −1
k −1

which is slightly different from the recurrence for the
Stirling numbers of the first kind. Here we have the
initial conditions:

n
n
¸
= 1, ∀n ∈ N and

n
0
¸
= 0, ∀n ≥ 1.
These relations completely define the Stirling trian-
gle of the second kind. Every row of the triangle
determines a polynomial; for example, from row 4 we
obtain: S
4
(w) = w +7w
2
+6w
3
+w
4
and it is called
the 4-th Stirling polynomial.
Let us now look for a combinatorial interpretation
of these numbers. If N
n
is the usual set ¦1, 2, . . . , n¦,
we can study the partitions of N
n
into k disjoint, non-
empty subsets. For example, when n = 4 and k = 2,
we have the following 7 partitions:
¦1¦ ∪ ¦2, 3, 4¦ ¦1, 2¦ ∪ ¦3, 4¦ ¦1, 3¦ ∪ ¦2, 4¦
¦1, 4¦ ∪ ¦2, 3¦ ¦1, 2, 3¦ ∪ ¦4¦
¦1, 2, 4¦ ∪ ¦3¦ ¦1, 3, 4¦ ∪ ¦2¦ .
If P
n,k
is the corresponding set, we now count [P
n,k
[
by fixing an element in N
n
, say the last element n.
The partitions in P
n,k
can contain n as a singleton
(i.e., as a subset with n as its only element) or can
contain n as an element in a larger subset. In the
former case, by eliminating ¦n¦ we obtain a partition
in P
n−1,k−1
and, obviously, all partitions in P
n−1,k−1
can be obtained in such a way. When n belongs to a
larger set, we can eliminate it obtaining a partition
in P
n−1,k
; however, the same partition is obtained
several times, exactly by eliminating n from any of
the k subsets containing it in the various partitions.
For example, the following three partitions in P
5,3
all
produce the same partition in P
4,3
:
¦1, 2, 5¦ ∪ ¦3¦ ∪ ¦4¦ ¦1, 2¦ ∪ ¦3, 5¦ ∪ ¦4¦
¦1, 2¦ ∪ ¦3¦ ∪ ¦4, 5¦
This proves the recurrence relation:
[P
n,k
[ = k[P
n−1,k
[ +[P
n−1,k−1
[
which is the same recurrence as for the Stirling num-
bers of the second kind. As far as the initial con-
ditions are concerned, we observe that there is only
one partition of N
n
composed by n subsets, i.e., the
partition containing n singletons; therefore [P
n,n
[ =
1, ∀n ∈ N (in the case n = 0 the empty set is the only
partition of the empty set). When n ≥ 1, there is
no partition of N
n
composed by 0 subsets, and there-
fore [P
n,0
[ = 0. We can conclude that [P
n,k
[ coincides
with the corresponding Stirling number of the second
kind, and use this fact for observing that:
•
¸
n
1
¸
= 1, ∀n ≥ 1. In fact, the only partition of
N
n
in a single subset is when the subset coincides
with N
n
;
•
¸
n
2
¸
= 2
n−1
, ∀n ≥ 2. When the partition is only
composed by two subsets, the first one uniquely
determines the second. Let us suppose that the
first subset always contains 1. By eliminating
1, we obtain as first subset all the subsets in
N
n
` ¦1¦, except this last whole set, which would
correspond to an empty second set. This proves
the identity;
•

n
n−1
¸
=

n
2

, ∀n ∈ N. In any partition with
one subset with 2 elements, and all the others
singletons, the two elements can be chosen in

n
2

different ways.
2.12 Bell and Bernoulli num-
bers
If we sum the rows of the Stirling triangle of the sec-
ond kind, we find a sequence:
n 0 1 2 3 4 5 6 7 8
B
n
1 1 2 5 15 52 203 877 4140
which represents the total number of partitions rela-
tive to the set N
n
. For example, the five partitions of
a set with three elements are:
¦1, 2, 3¦ ¦1¦ ∪ ¦2, 3¦ ¦1, 2¦ ∪ ¦3¦
24 CHAPTER 2. SPECIAL NUMBERS
¦1, 3¦ ∪ ¦2¦ ¦1¦ ∪ ¦2¦ ∪ ¦3¦ .
The numbers in this sequence are called Bell numbers
and are denoted by B
n
; by definition we have:
B
n
=
n
¸
k=0

n
k
¸
.
Bell numbers grow very fast; however, since
¸
n
k
¸
≤

n
k

, for every value of n and k (a subset in P
n,k
corresponds to one or more cycles in S
n,k
), we always
have B
n
≤ n!, and in fact B
n
< n! for every n > 1.
Another frequently occurring sequence is obtained
by ordering the subsets appearing in the partitions of
N
n
. For example, the partition ¦1¦∪¦2¦∪¦3¦ can be
ordered in 3! = 6 different ways, and ¦1, 2¦ ∪¦3¦ can
be ordered in 2! = 2 ways, i.e., ¦1, 2¦ ∪¦3¦ and ¦3¦ ∪
¦1, 2¦. These are called ordered partitions, and their
number is denoted by O
n
. By the previous example,
we easily see that O
3
= 13, and the sequence begins:
n 0 1 2 3 4 5 6 7
O
n
1 1 3 13 75 541 4683 47293
Because of this definition, the numbers O
n
are called
ordered Bell numbers and we have:
O
n
=
n
¸
k=0

n
k
¸
k!;
this shows that O
n
≥ n!, and, in fact, O
n
> n!, ∀n >
1.
Another combinatorial interpretation of the or-
dered Bell numbers is as follows. Let us fix an in-
teger n ∈ N and for every k ≤ n let A
k
be any mul-
tiset with n elements containing at least once all the
numbers 1, 2, . . . , k. The number of all the possible
orderings of the A
k
’s is just the nth ordered Bell num-
ber. For example, when n = 3, the possible multisets
are: ¦1, 1, 1¦ , ¦1, 1, 2¦ , ¦1, 2, 2¦ , ¦1, 2, 3¦. Their pos-
sible orderings are given by the following 7 vectors:
(1, 1, 1) (1, 1, 2) (1, 2, 1) (2, 1, 1)
(1, 2, 2) (2, 1, 2) (2, 2, 1)
plus the six permutations of the set ¦1, 2, 3¦. These
orderings are called preferential arrangements.
We can find a 1-1 correspondence between the
orderings of set partitions and preferential arrange-
ments. If (a
1
, a
2
, . . . , a
n
) is a preferential arrange-
ment, we build the corresponding ordered partition
by setting the element 1 in the a
1
th subset, 2 in the
a
2
th subset, and so on. If k is the largest number
in the arrangement, we exactly build k subsets. For
example, the partition corresponding to (1, 2, 2, 1) is
¦1, 4¦ ∪ ¦2, 3¦, while the partition corresponding to
(2, 1, 1, 2) is ¦2, 3¦∪¦1, 4¦, whose ordering is different.
This construction can be easily inverted and since it is
injective, we have proved that it is actually a 1-1 cor-
respondence. Because of that, ordered Bell numbers
are also called preferential arrangement numbers.
We conclude this section by introducing another
important sequence of numbers. These are (positive
or negative) rational numbers and therefore they can-
not correspond to any counting problem, i.e., their
combinatorial interpretation cannot be direct. How-
ever, they arise in many combinatorial problems and
therefore they should be examined here, for the mo-
ment only introducing their definition. The Bernoulli
numbers are implicitly defined by the recurrence re-
lation:
n
¸
k=0

n + 1
k

B
k
= δ
n,0
.
No initial condition is necessary, because for n = 0
we have

1
0

B
0
= 1, i.e., B
0
= 1. This is the starting
value, and B
1
is obtained by setting n = 1 in the
recurrence relation:

2
0

B
0
+

2
1

B
1
= 0.
We obtain B
1
= −1/2, and we now have a formula
for B
2
:

3
0

B
0
+

3
1

B
1
+

3
2

B
2
= 0.
By performing the necessary computations, we find
B
2
= 1/6, and we can go on successively obtaining
all the possible values for the B
n
’s. The first twelve
values are as follows:
n 0 1 2 3 4 5 6
B
n
1 −1/2 1/6 0 −1/30 0 1/42
n 7 8 9 10 11 12
B
n
0 −1/30 0 5/66 0 −691/2730
Except for B
1
, all the other values of B
n
for odd
n are zero. Initially, Bernoulli numbers seem to be
small, but as n grows, they become extremely large in
modulus, but, apart from the zero values, they are al-
ternatively one positive and one negative. These and
other properties of the Bernoulli numbers are not eas-
ily proven in a direct way, i.e., from their definition.
However, we’ll see later how we can arrange things
in such a way that everything becomes accessible to
us.
Chapter 3
Formal power series
3.1 Definitions for formal
power series
Let R be the field of real numbers and let t be any
indeterminate over R, i.e., a symbol different from
any element in R. A formal power series (f.p.s.) over
R in the indeterminate t is an expression:
f(t) = f
0
+f
1
t+f
2
t
2
+f
3
t
3
+ +f
n
t
n
+ =
∞
¸
k=0
f
k
t
k
where f
0
, f
1
, f
2
, . . . are all real numbers. The same
definition applies to every set of numbers, in particu-
lar to the field of rational numbers Q and to the field
of complex numbers C. The developments we are now
going to see, and depending on the field structure of
the numeric set, can be easily extended to every field
F of 0 characteristic. The set of formal power series
over F in the indeterminate t is denoted by F[[t]]. The
use of a particular indeterminate t is irrelevant, and
there exists an obvious 1-1 correspondence between,
say, F[[t]] and F[[y]]; it is simple to prove that this
correspondence is indeed an isomorphism. In order
to stress that our results are substantially indepen-
dent of the particular field F and of the particular
indeterminate t, we denote F[[t]] by T, but the reader
can think of T as of R[[t]]. In fact, in combinatorial
analysis and in the analysis of algorithms the coeffi-
cients f
0
, f
1
, f
2
, . . . of a formal power series are mostly
used to count objects, and therefore they are positive
integer numbers, or, in some cases, positive rational
numbers (e.g., when they are the coefficients of an ex-
ponential generating function. See below and Section
4.1).
If f(t) ∈ T, the order of f(t), denoted by ord(f(t)),
is the smallest index r for which f
r
= 0. The set of all
f.p.s. of order exactly r is denoted by T
r
or by F
r
[[t]].
The formal power series 0 = 0 + 0t + 0t
2
+ 0t
3
+
has infinite order.
If (f
0
, f
1
, f
2
, . . .) = (f
k
)
k∈N
is a sequence of (real)
numbers, there is no substantial difference between
the sequence and the f.p.s.
¸
∞
k=0
f
k
t
k
, which will
be called the (ordinary) generating function of the
sequence. The term ordinary is used to distinguish
these functions from exponential generating func-
tions, which will be introduced in the next chapter.
The indeterminate t is used as a “place-marker”, i.e.,
a symbol to denote the place of the element in the se-
quence. For example, in the f.p.s. 1+t +t
2
+t
3
+ ,
corresponding to the sequence (1, 1, 1, . . .), the term
t
5
= 1 t
5
simply denotes that the element in position
5 (starting from 0) in the sequence is the number 1.
Although our study of f.p.s. is mainly justified by
the development of a generating function theory, we
dedicate the present chapter to the general theory of
f.p.s., and postpone the study of generating functions
to the next chapter.
There are two main reasons why f.p.s. are more
easily studied than sequences:
1. the algebraic structure of f.p.s. is very well un-
derstood and can be developed in a standard
way;
2. many f.p.s. can be “abbreviated” by expressions
easily manipulated by elementary algebra.
The present chapter is devoted to these algebraic as-
pects of f.p.s.. For example, we will prove that the
series 1 +t +t
2
+t
3
+ can be conveniently abbre-
viated as 1/(1 −t), and from this fact we will be able
to infer that the series has a f.p.s. inverse, which is
1 −t + 0t
2
+ 0t
3
+ .
We conclude this section by defining the concept
of a formal Laurent (power) series (f.L.s.), as an ex-
pression:
g(t) = g
−m
t
−m
+g
−m+1
t
−m+1
+ +g
−1
t
−1
+
+g
0
+g
1
t +g
2
t
2
+ =
∞
¸
k=−m
g
k
t
k
.
The set of f.L.s. strictly contains the set of f.p.s..
For a f.L.s. g(t) the order can be negative; when the
order of g(t) is non-negative, then g(t) is actually a
f.p.s.. We observe explicitly that an expression as
¸
∞
k=−∞
f
k
t
k
does not represent a f.L.s..
25
26 CHAPTER 3. FORMAL POWER SERIES
3.2 The basic algebraic struc-
ture
The set T of f.p.s. can be embedded into several al-
gebraic structures. We are now going to define the
most common one, which is related to the usual con-
cept of sum and (Cauchy) product of series. Given
two f.p.s. f(t) =
¸
∞
k=0
f
k
t
k
and g(t) =
¸
∞
k=0
g
k
t
k
,
the sum of f(t) and g(t) is defined as:
f(t) +g(t) =
∞
¸
k=0
f
k
t
k
+
∞
¸
k=0
g
k
t
k
=
∞
¸
k=0
(f
k
+g
k
)t
k
.
From this definition, it immediately follows that T
is a commutative group with respect to the sum.
The associative and commutative laws directly fol-
low from the analogous properties in the field F; the
identity is the f.p.s. 0 = 0 + 0t + 0t
2
+ 0t
3
+ , and
the opposite series of f(t) =
¸
∞
k=0
f
k
t
k
is the series
−f(t) =
¸
∞
k=0
(−f
k
)t
k
.
Let us now define the Cauchy product of f(t) by
g(t):
f(t)g(t) =

∞
¸
k=0
f
k
t
k

∞
¸
k=0
g
k
t
k

=
=
∞
¸
k=0

¸
k
¸
j=0
f
j
g
k−j
¸

t
k
Because of the form of the t
k
coefficient, this is also
called the convolution of f(t) and g(t). It is a good
idea to write down explicitly the first terms of the
Cauchy product:
f(t)g(t) = f
0
g
0
+ (f
0
g
1
+f
1
g
0
)t +
+ (f
0
g
2
+f
1
g
1
+f
2
g
0
)t
2
+
+ (f
0
g
3
+f
1
g
2
+f
2
g
1
+f
3
g
0
)t
3
+
This clearly shows that the product is commutative
and it is a simple matter to prove that the identity is
the f.p.s. 1 = 1+0t +0t
2
+0t
3
+ . The distributive
law is a consequence of the distributive law valid in
F. In fact, we have:
(f(t) +g(t))h(t) =
=
∞
¸
k=0

¸
k
¸
j=0
(f
j
+g
j
)h
k−j
¸

t
k
=
=
∞
¸
k=0

¸
k
¸
j=0
f
j
h
k−j
¸

t
k
+
∞
¸
k=0

¸
k
¸
j=0
g
j
h
k−j
¸

t
k
=
= f(t)h(t) +g(t)h(t)
Finally, we can prove that T does not contain any
zero divisor. If f(t) and g(t) are two f.p.s. different
from zero, then we can suppose that ord(f(t)) = k
1
and ord(g(t)) = k
2
, with 0 ≤ k
1
, k
2
< ∞. This
means f
k1
= 0 and g
k2
= 0; therefore, the product
f(t)g(t) has the term of degree k
1
+k
2
with coefficient
f
k1
g
k2
= 0, and so it cannot be zero. We conclude
that (T, +, ) is an integrity domain.
The previous reasoning also shows that, in general,
we have:
ord(f(t)g(t)) = ord(f(t)) + ord(g(t))
The order of the identity 1 is obviously 0; if f(t) is an
invertible element in T, we should have f(t)f(t)
−1
=
1 and therefore ord(f(t)) = 0. On the other hand, if
f(t) ∈ T
0
, i.e., f(t) = f
0
+f
1
t +f
2
t
2
+f
3
t
3
+ with
f
0
= 0, we can easily prove that f(t) is invertible. In
fact, let g(t) = f(t)
−1
so that f(t)g(t) = 1. From
the explicit expression for the Cauchy product, we
can determine the coefficients of g(t) by solving the
infinite system of linear equations:







f
0
g
0
= 1
f
0
g
1
+f
1
g
0
= 0
f
0
g
2
+f
1
g
1
+f
2
g
0
= 0

The system can be solved in a simple way, starting
with the first equation and going on one equation
after the other. Explicitly, we obtain:
g
0
= f
−1
0
g
1
= −
f
1
f
2
0
g
2
= −
f
2
1
f
3
0
−
f
2
f
2
0

and therefore g(t) = f(t)
−1
is well defined. We con-
clude stating the result just obtained: a f.p.s. is in-
vertible if and only if its order is 0. Because of that,
T
0
is also called the set of invertible f.p.s.. Accord-
ing to standard terminology, the elements of T
0
are
called the units of the integrity domain.
As a simple example, let us compute the inverse of
the f.p.s. 1 −t = 1 −t +0t
2
+0t
3
+ . Here we have
f
0
= 1, f
1
= −1 and f
k
= 0, ∀k > 1. The system
becomes:






g
0
= 1
g
1
−g
0
= 0
g
2
−g
1
= 0

and we easily obtain that all the g
j
’s (j = 0, 1, 2, . . .)
are 1. Therefore the inverse f.p.s. we are looking for
is 1 + t + t
2
+ t
3
+ . The usual notation for this
fact is:
1
1 −t
= 1 +t +t
2
+t
3
+ .
It is well-known that this identity is only valid for
−1 < t < 1, when t is a variable and f.p.s. are inter-
preted as functions. In our formal approach, however,
these considerations are irrelevant and the identity is
valid from a purely formal point of view.
3.4. OPERATIONS ON FORMAL POWER SERIES 27
3.3 Formal Laurent Series
In the first section of this Chapter, we introduced the
concept of a formal Laurent series, as an extension
of the concept of a f.p.s.; if a(t) =
¸
∞
k=m
a
k
t
k
and
b(t) =
¸
∞
k=n
b
k
t
k
(m, n ∈ Z), are two f.L.s., we can
define the sum and the Cauchy product:
a(t) +b(t) =
∞
¸
k=m
a
k
t
k
+
∞
¸
k=n
b
k
t
k
=
=
∞
¸
k=p
(a
k
+b
k
)t
k
a(t)b(t) =

∞
¸
k=m
a
k
t
k

∞
¸
k=n
b
k
t
k

=
=
∞
¸
k=q

¸
¸
i+j=k
a
i
b
j
¸

t
k
where p = min(m, n) and q = m + n. As we did for
f.p.s., it is not difficult to find out that these opera-
tions enjoy the usual properties of sum and product,
and if we denote by L the set of f.L.s., we have that
(L, +, ) is a field. The only point we should formally
prove is that every f.L.s. a(t) =
¸
∞
k=m
a
k
t
k
= 0 has
an inverse f.L.s. b(t) =
¸
∞
k=−m
b
k
t
k
. However, this
is proved in the same way we proved that every f.p.s.
in T
0
has an inverse. In fact we should have:
a
m
b
−m
= 1
a
m
b
−m+1
+a
m+1
b
−m
= 0
a
m
b
−m+2
+a
m+1
b
−m+1
+a
m+2
b
−m
= 0

By solving the first equation, we find b
−m
= a
−1
m
;
then the system can be solved one equation after the
other, by substituting the values obtained up to the
moment. Since a
m
b
−m
is the coefficient of t
0
, we have
a(t)b(t) = 1 and the proof is complete.
We can now show that (L, +, ) is the smallest
field containing the integrity domain (T, +, ), thus
characterizing the set of f.L.s. in an algebraic way.
From Algebra we know that given an integrity do-
main (K, +, ) the smallest field (F, +, ) containing
(K, +, ) can be built in the following way: let us de-
fine an equivalence relation ∼ on the set K K:
(a, b) ∼ (c, d) ⇐⇒ ad = bc;
if we now set F = K K/ ∼, the set F with the
operations + and defined as the extension of + and
in K is the field we are searching for. This is just
the way in which the field Q of rational numbers is
constructed from the integrity domain Z of integers
numbers, and the field of rational functions is built
from the integrity domain of the polynomials.
Our aim is to show that the field (L, +, ) of f.L.s. is
isomorphic with the field constructed in the described
way starting with the integrity domain of f.p.s.. Let
´
L = T T be the set of pairs of f.p.s.; we begin
by showing that for every (f(t), g(t)) ∈
´
L there ex-
ists a pair (a(t), b(t)) ∈
´
L such that (f(t), g(t)) ∼
(a(t), b(t)) (i.e., f(t)b(t) = g(t)a(t)) and at least
one between a(t) and b(t) belongs to T
0
. In fact,
let p = min(ord(f(t)), ord(g(t))) and let us define
a(t), b(t) as f(t) = t
p
a(t) and g(t) = t
p
b(t); obviously,
either a(t) ∈ T
0
or b(t) ∈ T
0
or both are invertible
f.p.s.. We now have:
b(t)f(t) = b(t)t
p
a(t) = t
p
b(t)a(t) = g(t)a(t)
and this shows that (a(t), b(t)) ∼ (f(t), g(t)).
If b(t) ∈ T
0
, then a(t)/b(t) ∈ T and is uniquely de-
termined by a(t), b(t); in this case, therefore, our as-
sert is proved. So, let us now suppose that b(t) ∈ T
0
;
then we can write b(t) = t
m
v(t), where v(t) ∈ T
0
. We
have a(t)/v(t) =
¸
∞
k=0
d
k
t
k
∈ T
0
, and consequently
let us consider the f.L.s. l(t) =
¸
∞
k=0
d
k
t
k−m
; by
construction, it is uniquely determined by a(t), b(t)
or also by f(t), g(t). It is now easy to see that l(t)
is the inverse of the f.p.s. b(t)/a(t) in the sense of
f.L.s. as considered above, and our proof is complete.
This shows that the correspondence is a 1-1 corre-
spondence between L and
´
L preserving the inverse,
so it is now obvious that the correspondence is also
an isomorphism between (L, +, ) and (
´
L, +, ).
Because of this result, we can identify
´
L and L
and assert that (L, +, ) is indeed the smallest field
containing (T, +, ). From now on, the set
´
L will be
ignored and we will always refer to L as the field of
f.L.s..
3.4 Operations on formal
power series
Besides the four basic operations: addition, subtrac-
tion, multiplication and division, it is possible to con-
sider other operations on T, only a few of which can
be extended to L.
The most important operation is surely taking a
power of a f.p.s.; if p ∈ N we can recursively define:

f(t)
0
= 1 if p = 0
f(t)
p
= f(t)f(t)
p−1
if p > 0
and observe that ord(f(t)
p
) = p ord(f(t)). There-
fore, f(t)
p
∈ T
0
if and only if f(t) ∈ T
0
; on the
other hand, if f(t) ∈ T
0
, then the order of f(t)
p
be-
comes larger and larger and goes to ∞ when p → ∞.
This property will be important in our future devel-
opments, when we will reduce many operations to
28 CHAPTER 3. FORMAL POWER SERIES
infinite sums involving the powers f(t)
p
with p ∈ N.
If f(t) ∈ T
0
, i.e., ord(f(t)) > 0, these sums involve
elements of larger and larger order, and therefore for
every index k we can determine the coefficient of t
k
by only a finite number of terms. This assures that
our definitions will be good definitions.
We wish also to observe that taking a positive in-
teger power can be easily extended to L; in this case,
when ord(f(t)) < 0, ord(f(t)
p
) decreases, but re-
mains always finite. In particular, for g(t) = f(t)
−1
,
g(t)
p
= f(t)
−p
, and powers can be extended to all
integers p ∈ Z.
When the exponent p is a real or complex num-
ber whatsoever, we should restrict f(t)
p
to the case
f(t) ∈ T
0
; in fact, if f(t) = t
m
g(t), we would have:
f(t)
p
= (t
m
g(t))
p
= t
mp
g(t)
p
; however, t
mp
is an ex-
pression without any mathematical sense. Instead,
if f(t) ∈ T
0
, let us write f(t) = f
0
+ ´ v(t), with
ord(´ v(t)) > 0. For v(t) = ´ v(t)/f
0
, we have by New-
ton’s rule:
f(t)
p
= (f
0
+´ v(t))
p
= f
p
0
(1+v(t))
p
= f
p
0
∞
¸
k=0

p
k

v(t)
k
,
which can be assumed as a definition. In the last
expression, we can observe that: i) f
p
0
∈ C; ii)

p
k

is
defined for every value of p, k being a non-negative
integer; iii) v(t)
k
is well-defined by the considerations
above and ord(v(t)
k
) grows indefinitely, so that for
every k the coefficient of t
k
is obtained by a finite
sum. We can conclude that f(t)
p
is well-defined.
Particular cases are p = −1 and p = 1/2. In the
former case, f(t)
−1
is the inverse of the f.p.s. f(t).
We have already seen a method for computing f(t)
−1
,
but now we obtain the following formula:
f(t)
−1
=
1
f
0
∞
¸
k=0

−1
k

v(t)
k
=
1
f
0
∞
¸
k=0
(−1)
k
v(t)
k
.
For p = 1/2, we obtain a formula for the square root
of a f.p.s.:
f(t)
1/2
=

f(t) =

f
0
∞
¸
k=0

1/2
k

v(t)
k
=
=

f
0
∞
¸
k=0
(−1)
k−1
4
k
(2k −1)

2k
k

v(t)
k
.
In Section 3.12, we will see how f(t)
p
can be ob-
tained computationally without actually performing
the powers v(t)
k
. We conclude by observing that this
more general operation of taking the power p ∈ R
cannot be extended to f.L.s.: in fact, we would have
smaller and smaller terms t
k
(k → −∞) and there-
fore the resulting expression cannot be considered an
actual f.L.s., which requires a term with smallest de-
gree.
By applying well-known rules of the exponential
and logarithmic functions, we can easily define the
corresponding operations for f.p.s., which however,
as will be apparent, cannot be extended to f.L.s.. For
the exponentiation we have for f(t) ∈ T
0
, f(t) = f
0
+
v(t):
e
f(t)
= exp(f
0
+v(t)) = e
f0
∞
¸
k=0
v(t)
k
k!
.
Again, since v(t) ∈ T
0
, the order of v(t)
k
increases
with k and the sums necessary to compute the co-
efficient of t
k
are always finite. The formula makes
clear that exponentiation can be performed on every
f(t) ∈ T, and when f(t) ∈ T
0
the factor e
f0
is not
present.
For the logarithm, let us suppose f(t) ∈ T
0
, f(t) =
f
0
+ ´ v(t), v(t) = ´ v(t)/f
0
; then we have:
ln(f
0
+ ´ v(t)) = ln f
0
+ ln(1 +v(t)) =
= ln f
0
+
∞
¸
k=1
(−1)
k+1
v(t)
k
k
.
In this case, for f(t) ∈ T
0
, we cannot define the log-
arithm, and this shows an asymmetry between expo-
nential and logarithm.
Another important operation is differentiation:
Df(t) =
d
dt
f(t) =
∞
¸
k=1
kf
k
t
k−1
= f
′
(t).
This operation can be performed on every f(t) ∈ L,
and a very important observation is the following:
Theorem 3.4.1 For every f(t) ∈ L, its derivative
f
′
(t) does not contain any term in t
−1
.
Proof: In fact, by the general rule, the term in
t
−1
should have been originated by the constant term
(i.e., the term in t
0
) in f(t), but the product by k
reduces it to 0.
This fact will be the basis for very important results
on the theory of f.p.s. and f.L.s. (see Section 3.8).
Another operation is integration; because indefinite
integration leaves a constant term undefined, we pre-
fer to introduce and use only definite integration; for
f(t) ∈ T this is defined as:

t
0
f(τ)dτ =
∞
¸
k=0
f
k

t
0
τ
k
dτ =
∞
¸
k=0
f
k
k + 1
t
k+1
.
Our purely formal approach allows us to exchange
the integration and summation signs; in general, as
we know, this is only possible when the convergence is
uniform. By this definition,

t
0
f(τ)dτ never belongs
to T
0
. Integration can be extended to f.L.s. with an
3.6. COEFFICIENT EXTRACTION 29
obvious exception: because integration is the inverse
operation of differentiation, we cannot apply integra-
tion to a f.L.s. containing a term in t
−1
. Formally,
from the definition above, such a term would imply a
division by 0, and this is not allowed. In all the other
cases, integration does not create any problem.
3.5 Composition
A last operation on f.p.s. is so important that we
dedicate to it a complete section. The operation is the
composition of two f.p.s.. Let f(t) ∈ T and g(t) ∈ T
0
,
then we define the “composition” of f(t) by g(t) as
the f.p.s:
f(g(t)) =
∞
¸
k=0
f
k
g(t)
k
.
This definition justifies the fact that g(t) cannot be-
long to F
0
; in fact, otherwise, infinite sums were in-
volved in the computation of f(g(t)). In connection
with the composition of f.p.s., we will use the follow-
ing notation:
f(g(t)) =

f(y)


y = g(t)

which is intended to mean the result of substituting
g(t) to the indeterminate y in the f.p.s. f(y). The
indeterminate y is a dummy symbol; it should be dif-
ferent from t in order not to create any ambiguity,
but it can be substituted by any other symbol. Be-
cause every f(t) ∈ T
0
is characterized by the fact
that g(0) = 0, we will always understand that, in the
notation above, the f.p.s. g(t) is such that g(0) = 0.
The definition can be extended to every f(t) ∈ L,
but the function g(t) has always to be such that
ord(g(t)) > 0, otherwise the definition would imply
infinite sums, which we avoid because, by our formal
approach, we do not consider any convergence crite-
rion.
The f.p.s. in T
1
have a particular relevance for
composition. They are called quasi-units or delta se-
ries. First of all, we wish to observe that the f.p.s.
t ∈ T
1
acts as an identity for composition. In fact

y


y = g(t)

= g(t) and

f(y)


y = t

= f(t), and
therefore t is a left and right identity. As a sec-
ond fact, we show that a f.p.s. f(t) has an inverse
with respect to composition if and only if f(t) ∈ T
1
.
Note that g(t) is the inverse of f(t) if and only if
f(g(t)) = t and g(f(t)) = t. From this, we de-
duce immediately that f(t) ∈ T
0
and g(t) ∈ T
0
.
On the other hand, it is clear that ord(f(g(t))) =
ord(f(t))ord(g(t)) by our initial definition, and since
ord(t) = 1 and ord(f(t)) > 0, ord(g(t)) > 0, we must
have ord(f(t)) = ord(g(t)) = 1.
Let us now come to the main part of the proof
and consider the set T
1
with the operation of com-
position ◦; composition is always associative and
therefore (T
1
, ◦) is a group if we prove that every
f(t) ∈ T
1
has a left (or right) inverse, because the
theory assures that the other inverse exists and coin-
cides with the previously found inverse. Let f(t) =
f
1
t+f
2
t
2
+f
3
t
3
+ and g(t) = g
1
t+g
2
t
2
+g
3
t
3
+ ;
we have:
f(g(t)) = f
1
(g
1
t +g
2
t
2
+g
3
t
3
+ ) +
+f
2
(g
2
1
t
2
+ 2g
1
g
2
t
3
+ ) +
+f
3
(g
3
1
t
3
+ ) + =
= f
1
g
1
t + (f
1
g
2
+f
2
g
2
1
)t
2
+
+ (f
1
g
3
+ 2f
2
g
1
g
2
+f
3
g
3
1
)t
3
+
= t
In order to determine g(t) we have to solve the sys-
tem:







f
1
g
1
= 1
f
1
g
2
+f
2
g
2
1
= 0
f
1
g
3
+ 2f
2
g
1
g
2
+f
3
g
3
1
= 0

The first equation gives g
1
= 1/f
1
; we can substitute
this value in the second equation and obtain a value
for g
2
; the two values for g
1
and g
2
can be substi-
tuted in the third equation and obtain a value for g
3
.
Continuing in this way, we obtain the value of all the
coefficients of g(t), and therefore g(t) is determined
in a unique way. In fact, we observe that, by con-
struction, in the kth equation, g
k
appears in linear
form and its coefficient is always f
1
. Being f
1
= 0,
g
k
is unique even if the other g
r
(r < k) appear with
powers.
The f.p.s. g(t) such that f(g(t)) = t, and therefore
such that g(f(t)) = t as well, is called the composi-
tional inverse of f(t). In the literature, it is usually
denoted by f(t) or f
[−1]
(t); we will adopt the first no-
tation. Obviously, f(t) = f(t), and sometimes f(t) is
also called the reverse of f(t). Given f(t) ∈ T
1
, the
determination of its compositional inverse is one of
the most interesting problems in the theory of f.p.s.
or f.L.s.; it was solved by Lagrange and we will dis-
cuss it in the following sections. Note that, in princi-
ple, the g
k
’s can be computed by solving the system
above; this, however, is too complicated and nobody
will follow that way, unless for exercising.
3.6 Coefficient extraction
If f(t) ∈ L, or in particular f(t) ∈ T, the notation
[t
n
]f(t) indicates the extraction of the coefficient of
t
n
from f(t), and therefore we have [t
n
]f(t) = f
n
.
In this sense, [t
n
] can be seen as a mapping: [t
n
] :
L → R or [t
n
] : L → C, according to what is the
field underlying the set L or T. Because of that, [t
n
]
30 CHAPTER 3. FORMAL POWER SERIES
(linearity) [t
n
](αf(t) +βg(t)) = α[t
n
]f(t) +β[t
n
]g(t) (K1)
(shifting) [t
n
]tf(t) = [t
n−1
]f(t) (K2)
(differentiation) [t
n
]f
′
(t) = (n + 1)[t
n+1
]f(t) (K3)
(convolution) [t
n
]f(t)g(t) =
n
¸
k=0
[t
k
]f(t)[t
n−k
]g(t) (K4)
(composition) [t
n
]f(g(t)) =
∞
¸
k=0
([y
k
]f(y))[t
n
]g(t)
k
(K5)
Table 3.1: The rules for coefficient extraction
is called an operator and exactly the “coefficient of ”
operator or, more simply, the coefficient operator.
In Table 3.1 we state formally the main properties
of this operator, by collecting what we said in the pre-
vious sections. We observe that α, β ∈ R or α, β ∈ C
are any constants; the use of the indeterminate y is
only necessary not to confuse the action on different
f.p.s.; because g(0) = 0 in composition, the last sum
is actually finite. Some points require more lengthy
comments. The property of shifting can be easily gen-
eralized to [t
n
]t
k
f(t) = [t
n−k
]f(t) and also to nega-
tive powers: [t
n
]f(t)/t
k
= [t
n+k
]f(t). These rules are
very important and are often applied in the theory of
f.p.s. and f.L.s.. In the former case, some care should
be exercised to see whether the properties remain in
the realm of T or go beyond it, invading the domain
of L, which can be not always correct. The property
of differentiation for n = 1 gives [t
−1
]f
′
(t) = 0, a
situation we already noticed. The operator [t
−1
] is
also called the residue and is noted as “res”; so, for
example, people write resf
′
(t) = 0 and some authors
use the notation res t
n+1
f(t) for [t
n
]f(t).
We will have many occasions to apply rules (K1)÷
(K5) of coefficient extraction. However, just to give
a meaningful example, let us find the coefficient of
t
n
in the series expansion of (1 + αt)
r
, when α and
r are two real numbers whatsoever. Rule (K3) can
be written in the form [t
n
]f(t) =
1
n
[t
n−1
]f
′
(t) and we
can successively apply this form to our case:
[t
n
](1 +αt)
r
=
rα
n
[t
n−1
](1 +αt)
r−1
=
=
rα
n
(r −1)α
n −1
[t
n−2
](1 +αt)
r−2
= =
=
rα
n
(r −1)α
n −1

(r −n + 1)α
1
[t
0
](1 +αt)
r−n
=
= α
n

r
n

[t
0
](1 +αt)
r−n
.
We now observe that [t
0
](1 + αt)
r−n
= 1 because of
our observations on f.p.s. operations. Therefore, we
conclude with the so-called Newton’s rule:
[t
n
](1 +αt)
r
=

r
n

α
n
which is one of the most frequently used results in
coefficient extraction. Let us remark explicitly that
when r = −1 (the geometric series) we have:
[t
n
]
1
1 +αt
=

−1
n

α
n
=
=

1 +n −1
n

(−1)
n
α
n
= (−α)
n
.
A simple, but important use of Newton’s rule con-
cerns the extraction of the coefficient of t
n
from the
inverse of a trinomial at
2
+ bt + c, in the case it is
reducible, i.e., it can be written (1 +αt)(1 +βt); ob-
viously, we can always reduce the constant c to 1; by
the linearity rule, it can be taken outside the “coeffi-
cient of” operator. Therefore, our aim is to compute:
[t
n
]
1
(1 +αt)(1 +βt)
with α = β, otherwise Newton’s rule would be im-
mediately applicable. The problem can be solved by
using the technique of partial fraction expansion. We
look for two constants A and B such that:
1
(1 +αt)(1 +βt)
=
A
1 +αt
+
B
1 +βt
=
=
A+Aβt +B +Bαt
(1 +αt)(1 +βt)
;
if two such constants exist, the numerator in the first
expression should equal the numerator in the last one,
independently of t, or, if one so prefers, for every
value of t. Therefore, the term A+B should be equal
to 1, while the term (Aβ +Bα)t should always be 0.
The values for A and B are therefore the solution of
the linear system:

A+B = 1
Aβ +Bα = 0
3.7. MATRIX REPRESENTATION 31
The discriminant of this system is α − β, which is
always different from 0, because of our hypothesis
α = β. The system has therefore only one solution,
which is A = α/(α−β) and B = −β/(α−β). We can
now substitute these values in the expression above:
[t
n
]
1
(1 +αt)(1 +βt)
=
= [t
n
]
1
α −β

α
1 +αt
−
β
1 +βt

=
=
1
α −β

[t
n
]
α
1 +αt
−[t
n
]
β
1 +βt

=
=
α
n+1
−β
n+1
α −β
(−1)
n
Let us now consider a trinomial 1 + bt + ct
2
for
which ∆ = b
2
− 4c < 0 and b = 0. The trinomial is
irreducible, but we can write:
[t
n
]
1
1 +bt +ct
2
=
= [t
n
]
1

1 −
−b+i
√
|∆|
2
t

1 −
−b−i
√
|∆|
2
t

This time, a partial fraction expansion does not give
a simple closed form for the coefficients, however, we
can apply the formula above in the form:
[t
n
]
1
(1 −αt)(1 −βt)
=
α
n+1
−β
n+1
α −β
.
Since α and β are complex numbers, the resulting
expression is not very appealing. We can try to give
it a better form. Let us set α =

−b +i

[∆[

/2,
so α is always contained in the positive imaginary
halfplane. This implies 0 < arg(α) < π and we have:
α −β = −
b
2
+i

[∆[
2
+
b
2
+i

[∆[
2
=
= i

[∆[ = i

4c −b
2
If θ = arg(α) and:
ρ = [α[ =

b
2
4
−
4c −b
2
4
=
√
c
we can set α = ρe
iθ
and β = ρe
−iθ
. Consequently:
α
n+1
−β
n+1
= ρ
n+1

e
i(n+1)θ
−e
−i(n+1)θ

=
= 2iρ
n+1
sin(n + 1)θ
and therefore:
[t
n
]
1
1 +bt +ct
2
=
2(
√
c)
n+1
sin(n + 1)θ
√
4c −b
2
At this point we only have to find the value of θ.
Obviously:
θ = arctan


[∆[
2

−b
2

+kπ = arctan
√
4c −b
2
−b
+kπ
When b < 0, we have 0 < arctan

−
√
4c −b
2

/2 <
π/2, and this is the correct value for θ. However,
when b > 0, the principal branch of arctan is negative,
and we should set θ = π+arctan

−
√
4c −b
2

/2. As
a consequence, we have:
θ = arctan
√
4c −b
2
−b
+C
where C = π if b > 0 and C = 0 if b < 0.
An interesting and non-trivial example is given by:
σ
n
= [t
n
]
1
1 −3t + 3t
2
=
=
2(
√
3)
n+1
sin((n + 1) arctan(
√
3/3))
√
3
=
= 2(
√
3)
n
sin
(n + 1)π
6
These coefficients have the following values:
n = 12k σ
n
=
√
3
12k
= 729
k
n = 12k + 1 σ
n
=
√
3
12k+2
= 3 729
k
n = 12k + 2 σ
n
= 2
√
3
12k+2
= 6 729
k
n = 12k + 3 σ
n
=
√
3
12k+4
= 9 729
k
n = 12k + 4 σ
n
=
√
3
12k+4
= 9 729
k
n = 12k + 5 σ
n
= 0
n = 12k + 6 σ
n
= −
√
3
12k+6
= −27 729
k
n = 12k + 7 σ
n
= −
√
3
12k+8
= −81 729
k
n = 12k + 8 σ
n
= −2
√
3
12k+8
= −162 729
k
n = 12k + 9 σ
n
= −
√
3
12k+10
= −243 729
k
n = 12k + 10 σ
n
= −
√
3
12k+10
= −243 729
k
n = 12k + 11 σ
n
= 0
3.7 Matrix representation
Let f(t) ∈ T
0
; with the coefficients of f(t) we form
the following infinite lower triangular matrix (or ar-
ray) D = (d
n,k
)
n,k∈N
: column 0 contains the coeffi-
cients f
0
, f
1
, f
2
, . . . in this order; column 1 contains
the same coefficients shifted down by one position
and d
0,1
= 0; in general, column k contains the coef-
ficients of f(t) shifted down k positions, so that the
first k positions are 0. This definition can be sum-
marized in the formula d
n,k
= f
n−k
, ∀n, k ∈ N. For
a reason which will be apparent only later, the array
32 CHAPTER 3. FORMAL POWER SERIES
D will be denoted by (f(t), 1):
D = (f(t), 1) =

¸
¸
¸
¸
¸
¸
¸
¸
f
0
0 0 0 0
f
1
f
0
0 0 0
f
2
f
1
f
0
0 0
f
3
f
2
f
1
f
0
0
f
4
f
3
f
2
f
1
f
0

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
¸








.
If (f(t), 1) and (g(t), 1) are the matrices corre-
sponding to the two f.p.s. f(t) and g(t), we are in-
terested in finding out what is the matrix obtained
by multiplying the two matrices with the usual row-
by-column product. This product will be denoted by
(f(t), 1) (g(t), 1), and it is immediate to see what
its generic element d
n,k
is. The row n in (f(t), 1) is,
by definition, ¦f
n
, f
n−1
, f
n−2
, . . .¦, and column k in
(g(t), 1) is ¦0, 0, . . . , 0, g
1
, g
2
, . . .¦ where the number
of leading 0’s is just k. Therefore we have:
d
n,k
=
∞
¸
j=0
f
n−j
g
j−k
if we conventionally set g
r
= 0, ∀r < 0, When
k = 0, we have d
n,0
=
¸
∞
j=0
f
n−j
g
j
=
¸
n
j=0
f
n−j
g
j
,
and therefore column 0 contains the coefficients of
the convolution f(t)g(t). When k = 1 we have
d
n,1
=
¸
∞
j=0
f
n−j
g
j−1
=
¸
n−1
j=0
f
n−1−j
g
j
, and this
is the coefficient of t
n−1
in the convolution f(t)g(t).
Proceeding in the same way, we see that column k
contains the coefficients of the convolution f(t)g(t)
shifted down k positions. Therefore we conclude:
(f(t), 1) (g(t), 1) = (f(t)g(t), 1)
and this shows that there exists a group isomorphism
between (T
0
, ) and the set of matrices (f(t), 1) with
the row-by-column product. In particular, (1, 1) is
the identity (in fact, it corresponds to the identity
matrix) and (f(t)
−1
, 1) is the inverse of (f(t), 1).
Let us now consider a f.p.s. f(t) ∈ T
1
and let
us build an infinite lower triangular matrix in the
following way: column k contains the coefficients of
f(t)
k
in their proper order:

¸
¸
¸
¸
¸
¸
¸
¸
1 0 0 0 0
0 f
1
0 0 0
0 f
2
f
2
1
0 0
0 f
3
2f
1
f
2
f
3
1
0
0 f
4
2f
1
f
3
+f
2
2
3f
2
1
f
2
f
4
1

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
¸








.
The matrix will be denoted by (1, f(t)/t) and we are
interested to see how the matrix (1, g(t)/t)(1, f(t)/t)
is composed when f(t), g(t) ∈ T
1
.
If

´
f
n,k

n,k∈N
= (1, f(t)/t), by definition we have:
´
f
n,k
= [t
n
]f(t)
k
and therefore the generic element d
n,k
of the product
is:
d
n,k
=
∞
¸
j=0
´ g
n,j
´
f
j,k
=
∞
¸
j=0
[t
n
]g(t)
j
[y
j
]f(y)
k
=
= [t
n
]
∞
¸
j=0

[y
j
]f(t)
k

g(t)
j
= [t
n
]f(g(t))
k
.
In other words, column k in (1, g(t)/t) (1, f(t)/t) is
the kth power of the composition f(g(t)), and we can
conclude:
(1, g(t)/t) (1, f(t)/t) = (1, f(g(t))/t).
Clearly, the identity t ∈ T
1
corresponds to the ma-
trix (1, t/t) = (1, 1), the identity matrix, and this is
sufficient to prove that the correspondence f(t) ↔
(1, f(t)/t) is a group isomorphism.
Row-by-column product is surely the basic op-
eration on matrices and its extension to infinite,
lower triangular arrays is straight-forward, because
the sums involved in the product are actually finite.
We have shown that we can associate every f.p.s.
f(t) ∈ T
0
to a particular matrix (f(t), 1) (let us
denote by A the set of such arrays) in such a way
that (T
0
, ) is isomorphic to (A, ), and the Cauchy
product becomes the row-by-column product. Be-
sides, we can associate every f.p.s. g(t) ∈ T
1
to a
matrix (1, g(t)/t) (let us call B the set of such matri-
ces) in such a way that (T
1
, ◦) is isomorphic to (B, ),
and the composition of f.p.s. becomes again the row-
by-column product. This reveals a connection be-
tween the Cauchy product and the composition: in
the Chapter on Riordan Arrays we will explore more
deeply this connection; for the moment, we wish to
see how this observation yields to a computational
method for evaluating the compositional inverse of a
f.p.s. in T
1
.
3.8 Lagrange inversion theo-
rem
Given an infinite, lower triangular array of the form
(1, f(t)/t), with f(t) ∈ T
1
, the inverse matrix
(1, g(t)/t) is such that (1, g(t)/t) (1, f(t)/t) = (1, 1),
and since the product results in (1, f(g(t))/t) we have
f(g(t)) = t. In other words, because of the isomor-
phism we have seen, the inverse matrix for (1, f(t)/t)
is just the matrix corresponding to the compositional
inverse of f(t). As we have already said, Lagrange
3.9. SOME EXAMPLES OF THE LIF 33
found a noteworthy formula for the coefficients of
this compositional inverse. We follow the more recent
proof of Stanley, which points out the purely formal
aspects of Lagrange’s formula. Indeed, we will prove
something more, by finding the exact form of the ma-
trix (1, g(t)/t), inverse of (1, f(t)/t). As a matter of
fact, we state what the form of (1, g(t)/t) should be
and then verify that it is actually so.
Let D = (d
n,k
)
n,k∈N
be defined as:
d
n,k
=
k
n
[t
n−k
]

t
f(t)

n
.
Because f(t)/t ∈ T
0
, the power (t/f(t))
k
=
(f(t)/t)
−k
is well-defined; in order to show that
(d
n,k
)
n,k∈N
= (1, g(t)/t) we only have to prove that
D (1, f(t)/t) = (1, 1), because we already know that
the compositional inverse of f(t) is unique. The
generic element v
n,k
of the row-by-column product
D (1, f(t)/t) is:
v
n,k
=
∞
¸
j=0
d
n,j
[y
j
]f(y)
k
=
=
∞
¸
j=0
j
n
[t
n−j
]

t
f(t)

n
[y
j
]f(y)
k
.
By the rule of differentiation for the coefficient of op-
erator, we have:
j[y
j
]f(y)
k
= [y
j−1
]
d
dy
f(y)
k
= k[y
j
]yf
′
(y)f(y)
k−1
.
Therefore, for v
n,k
we have:
v
n,k
=
k
n
∞
¸
j=0
[t
n−j
]

t
f(t)

n
[y
j
]yf
′
(y)f(y)
k−1
=
=
k
n
[t
n
]

t
f(t)

n
tf
′
(t)f(t)
k−1
.
In fact, the factor k/n does not depend on j and
can be taken out of the summation sign; the sum
is actually finite and is the term of the convolution
appearing in the last formula. Let us now distinguish
between the case k = n and k = n. When k = n we
have:
v
n,n
= [t
n
]t
n
tf(t)
−1
f
′
(t) =
= [t
0
]f
′
(t)

t
f(t)

= 1;
in fact, f
′
(t) = f
1
+ 2f
2
t + 3f
3
t
2
+ and, being
f(t)/t ∈ T
0
, (f(t)/t)
−1
= (f
1
+f
2
t +f
3
t
2
+ )
−1
=
f
−1
1
+ ; therefore, the constant term in f
′
(t)(t/f(t))
is f
1
/f
1
= 1. When k = n:
v
n,k
=
k
n
[t
n
]t
n
tf(t)
k−n−1
f
′
(t) =
=
k
n
[t
−1
]
1
k −n
d
dt

f(t)
k−n

= 0;
in fact, f(t)
k−n
is a f.L.s. and, as we observed, the
residue of its derivative should be zero. This proves
that D (1, f(t)/t) = (1, 1) and therefore D is the
inverse of (1, f(t)/t).
If f(t) is the compositional inverse of f(t), the col-
umn 1 gives us the value of its coefficients; by the
formula for d
n,k
we have:
f
n
= [t
n
]f(t) = d
n,1
=
1
n
[t
n−1
]

t
f(t)

n
and this is the celebrated Lagrange Inversion For-
mula (LIF). The other columns give us the coefficients
of the powers f(t)
k
, for which we have:
[t
n
]f(t)
k
=
k
n
[t
n−k
]

t
f(t)

n
.
Many times, there is another way for applying the
LIF. Suppose we have a functional equation w =
tφ(w), where φ(t) ∈ T
0
, and we wish to find the
f.p.s. w = w(t) satisfying this functional equation.
Clearly w(t) ∈ T
1
and if we set f(y) = y/φ(y), we
also have f(t) ∈ T
1
. However, the functional equa-
tion can be written f(w(t)) = t, and this shows that
w(t) is the compositional inverse of f(t). We there-
fore know that w(t) is uniquely determined and the
LIF gives us:
[t
n
]w(t) =
1
n
[t
n−1
]

t
f(t)

n
=
1
n
[t
n−1
]φ(t)
n
.
The LIF can also give us the coefficients of the
powers w(t)
k
, but we can obtain a still more general
result. Let F(t) ∈ T and let us consider the com-
position F(w(t)) where w = w(t) is, as before, the
solution to the functional equation w = tφ(w), with
φ(w) ∈ T
0
. For the coefficient of t
n
in F(w(t)) we
have:
[t
n
]F(w(t)) = [t
n
]
∞
¸
k=0
F
k
w(t)
k
=
=
∞
¸
k=0
F
k
[t
n
]w(t)
k
=
∞
¸
k=0
F
k
k
n
[t
n−k
]φ(t)
n
=
=
1
n
[t
n−1
]

∞
¸
k=0
kF
k
t
k−1

φ(t)
n
=
=
1
n
[t
n−1
]F
′
(t)φ(t)
n
.
Note that [t
0
]F(w(t)) = F
0
, and this formula can be
generalized to every F(t) ∈ L, except for the coeffi-
cient [t
0
]F(w(t)).
3.9 Some examples of the LIF
We found that the number b
n
of binary trees
with n nodes (and of other combinatorial objects
34 CHAPTER 3. FORMAL POWER SERIES
as well) satisfies the recurrence relation: b
n+1
=
¸
n
k=0
b
k
b
n−k
. Let us consider the f.p.s. b(t) =
¸
∞
k=0
b
k
t
k
; if we multiply the recurrence relation by
t
n+1
and sum for n from 0 to infinity, we find:
∞
¸
n=0
b
n+1
t
n+1
=
∞
¸
n=0
t
n+1

n
¸
k=0
b
k
b
n−k

.
Since b
0
= 1, we can add and subtract 1 = b
0
t
0
from
the left hand member and can take t outside the sum-
mation sign in the right hand member:
∞
¸
n=0
b
n
t
n
−1 = t
∞
¸
n=0

n
¸
k=0
b
k
b
n−k

t
n
.
In the r.h.s. we recognize a convolution and substi-
tuting b(t) for the corresponding f.p.s., we obtain:
b(t) −1 = tb(t)
2
.
We are interested in evaluating b
n
= [t
n
]b(t); let us
therefore set w = w(t) = b(t) −1, so that w(t) ∈ T
1
and w
n
= b
n
, ∀n > 0. The previous relation becomes
w = t(1+w)
2
and we see that the LIF can be applied
(in the form relative to the functional equation) with
φ(t) = (1 +t)
2
. Therefore we have:
b
n
= [t
n
]w(t) =
1
n
[t
n−1
](1 +t)
2n
=
=
1
n

2n
n −1

=
1
n
(2n)!
(n −1)!(n + 1)!
=
=
1
n + 1
(2n)!
n!n!
=
1
n + 1

2n
n

.
As we said in the previous chapter, b
n
is called the
nth Catalan number and, under this name, it is often
denoted by C
n
. Now we have its form:
C
n
=
1
n + 1

2n
n

also valid in the case n = 0 when C
0
= 1.
In the same way we can compute the number of
p-ary trees with n nodes. A p-ary tree is a tree in
which all the nodes have arity p, except for leaves,
which have arity 0. A non-empty p-ary tree can be
decomposed in the following way:
,
, , , ¨
¨
¨
¨
¨
¨
¨
¨¨




r
r
r
r
r
r
r
r r



t
t
tt
T
1



t
t
tt
T
2
. . .



t
t
tt
T
p
which proves that T
n+1
=
¸
T
i1
T
i2
T
ip
, where the
sum is extended to all the p-uples (i
1
, i
2
, . . . , i
p
) such
that i
1
+i
2
+ +i
p
= n. As before, we can multiply
by t
n+1
the two members of the recurrence relation
and sum for n from 0 to infinity. We find:
T(t) −1 = tT(t)
p
.
This time we have a p degree equation, which cannot
be directly solved. However, if we set w(t) = T(t)−1,
so that w(t) ∈ T
1
, we have:
w = t(1 +w)
p
and the LIF gives:
T
n
= [t
n
]w(t) =
1
n
[t
n−1
](1 +t)
pn
=
=
1
n

pn
n −1

=
1
n
(pn)!
(n −1)!((p −1)n + 1)!
=
=
1
(p −1)n + 1

pn
n

which generalizes the formula for the Catalan num-
bers.
Finally, let us find the solution of the functional
equation w = te
w
. The LIF gives:
w
n
=
1
n
[t
n−1
]e
nt
=
1
n
n
n−1
(n −1)!
=
n
n−1
n!
.
Therefore, the solution we are looking for is the f.p.s.:
w(t) =
∞
¸
n=1
n
n−1
n!
t
n
= t +t
2
+
3
2
t
3
+
8
3
t
4
+
125
24
t
5
+ .
As noticed, w(t) is the compositional inverse of
f(t) = t/φ(t) = te
−t
= t −t
2
+
t
3
2!
−
t
4
3!
+
t
5
4!
− .
It is a useful exercise to perform the necessary com-
putations to show that f(w(t)) = t, for example up to
the term of degree 5 or 6, and verify that w(f(t)) = t
as well.
3.10 Formal power series and
the computer
When we are dealing with generating functions, or
more in general with formal power series of any kind,
we often have to perform numerical computations in
order to verify some theoretical result or to experi-
ment with actual cases. In these and other circum-
stances the computer can help very much with its
speed and precision. Nowadays, several Computer
Algebra Systems exist, which offer the possibility of
3.11. THE INTERNAL REPRESENTATION OF EXPRESSIONS 35
actually working with formal power series, contain-
ing formal parameters as well. The use of these tools
is recommended because they can solve a doubt in
a few seconds, can clarify difficult theoretical points
and can give useful hints whenever we are faced with
particular problems.
However, a Computer Algebra System is not al-
ways accessible or, in certain circumstances, one may
desire to use less sophisticated tools. For example,
programmable pocket computers are now available,
which can perform quite easily the basic operations
on formal power series. The aim of the present and
of the following sections is to describe the main al-
gorithms for dealing with formal power series. They
can be used to program a computer or to simply un-
derstand how an existing system actually works.
The simplest way to represent a formal power se-
ries is surely by means of a vector, in which the kth
component (starting from 0) is the coefficient of t
k
in
the power series. Obviously, the computer memory
only can store a finite number of components, so an
upper bound n
0
is usually given to the length of vec-
tors and to represent power series. In other words we
have:
repr
n0

∞
¸
k=0
a
k
t
k

= (a
0
, a
1
, . . . , a
n
) (n ≤ n
0
)
Fortunately, most operations on formal power se-
ries preserve the number of significant components,
so that there is little danger that a number of succes-
sive operations could reduce a finite representation to
a meaningless sequence of numbers. Differentiation
decreases by one the number of useful components;
on the contrary, integration and multiplication by t
r
,
say, increase the number of significant elements, at
the cost of introducing some 0 components.
The components a
0
, a
1
, . . . , a
n
are usually real
numbers, represented with the precision allowed by
the particular computer. In most combinatorial ap-
plications, however, a
0
, a
1
, . . . , a
n
are rational num-
bers and, with some extra effort, it is not difficult
to realize rational arithmetic on a computer. It is
sufficient to represent a rational number as a couple
(m, n), whose intended meaning is just m/n. So we
must have m ∈ Z, n ∈ N and it is a good idea to
keep m and n coprime. This can be performed by
a routine reduce computing p = gcd(m, n) using Eu-
clid’s algorithm and then dividing both m and n by
p. The operations on rational numbers are defined in
the following way:
(m, n) + (m
′
, n
′
) = reduce(mn
′
+m
′
n, nn
′
)
(m, n) (m
′
, n
′
) = reduce(mm
′
, nn
′
)
(m, n)
−1
= (n, m)
(m, n)
p
= (m
p
, n
p
) (p ∈ N)
provided that (m, n) is a reduced rational number.
The dimension of m and n is limited by the internal
representation of integer numbers.
In order to avoid this last problem, Computer Al-
gebra Systems usually realize an indefinite precision
integer arithmetic. An integer number has a vari-
able length internal representation and special rou-
tines are used to perform the basic operations. These
routines can also be realized in a high level program-
ming language (such as C or JAVA), but they can
slow down too much execution time if realized on a
programmable pocket computer.
3.11 The internal representa-
tion of expressions
The simple representation of a formal power series by
a vector of real, or rational, components will be used
in the next sections to explain the main algorithms
for formal power series operations. However, it is
surely not the best way to represent power series and
becomes completely useless when, for example, the
coefficients depend on some formal parameter. In
other words, our representation only can deal with
purely numerical formal power series.
Because of that, Computer Algebra Systems use a
more sophisticated internal representation. In fact,
power series are simply a particular case of a general
mathematical expression. The aim of the present sec-
tion is to give a rough idea of how an expression can
be represented in the computer memory.
In general, an expression consists in operators and
operands. For example, in a +
√
3, the operators are
+ and
√
, and the operands are a and 3. Every
operator has its own arity or adicity, i.e., the number
of operands on which it acts. The adicity of the sum
+ is two, because it acts on its two terms; the adicity
of the square root
√
is one, because it acts on a
single term. Operands can be numbers (and it is
important that the nature of the number be specified,
i.e., if it is a natural, an integer, a rational, a real or a
complex number) or can be a formal parameter, as a.
Obviously, if an operator acts on numerical operands,
it can be executed giving a numerical result. But if
any of its operands is a formal parameter, the result is
a formal expression, which may perhaps be simplified
but cannot be evaluated to a numerical result.
An expression can always be transposed into a
“tree”, the internal nodes of which correspond to
operators and whose leaves correspond to operands.
The simple tree for the previous expression is given
in Figure 3.1. Each operator has as many branches as
its adicity is and a simple visit to the tree can perform
its evaluation, that is it can execute all the operators
36 CHAPTER 3. FORMAL POWER SERIES
+
√
a
`

3
`




d
d
d
Figure 3.1: The tree for a simple expression
when only numerical operands are attached to them.
Simplification is a rather complicated matter and it
is not quite clear what a “simple” expression is. For
example, which is simpler between (a+1)(a+2) and
a
2
+3a+2? It is easily seen that there are occasions in
which either expression can be considered “simpler”.
Therefore, most Computer Algebra Systems provide
both a general “simplification” routine together with
a series of more specific programs performing some
specific simplifying tasks, such as expanding paren-
thetized expressions or collecting like factors.
In the computer memory an expression is repre-
sented by its tree, which is called the tree or list repre-
sentation of the expression. The representation of the
formal power series
¸
n
k=0
a
k
t
k
is shown in Figure 3.2.
This representation is very convenient and when some
coefficients a
1
, a
2
, . . . , a
n
depend on a formal param-
eter p nothing is changed, at least conceptually. In
fact, where we have drawn a leaf a
j
, we simply have
a more complex tree representing the expression for
a
j
.
The reader can develop computer programs for
dealing with this representation of formal power se-
ries. It should be clear that another important point
of this approach is that no limitation is given to the
length of expressions. A clever and dynamic use of
the storage solves every problem without increasing
the complexity of the corresponding programs.
3.12 Basic operations of formal
power series
We are now considering the vector representation of
formal power series:
repr
n0

∞
¸
k=0
a
k
t
k

= (a
0
, a
1
, . . . , a
n
) n ≤ n
0
.
The sum of the formal power series is defined in an
obvious way:
(a
0
, a
1
, . . . , a
n
) + (b
0
, b
1
, . . . , b
m
) = (c
0
, c
1
, . . . , c
r
)
where c
i
= a
i
+ b
i
for every 0 ≤ i ≤ r, and r =
min(n, m). In a similar way the Cauchy product is
defined:
(a
0
, a
1
, . . . , a
n
) (b
0
, b
1
, . . . , b
m
) = (c
0
, c
1
, . . . , c
r
)
where c
k
=
¸
k
j=0
a
j
b
k−j
for every 0 ≤ k ≤ r. Here
r is defined as r = min(n + p
B
, m + p
A
), if p
A
is
the first index for which a
pA
= 0 and p
B
is the first
index for which b
pB
= 0. We point out that the
time complexity for the sum is O(r) and the time
complexity for the product is O(r
2
).
Subtraction is similar to addition and does not re-
quire any particular comment. Before discussing divi-
sion, let us consider the operation of rising a formal
power series to a power α ∈ R. This includes the
inversion of a power series (α = −1) and therefore
division as well.
First of all we observe that whenever α ∈ N, f(t)
α
can be reduced to that case (1 +g(t))
α
, where g(t) / ∈
T
0
. In fact we have:
f(t) = f
h
t
h
+f
h+1
t
h+1
+f
h+2
t
h+2
+ =
= f
h
t
h

1 +
f
h+1
f
h
t +
f
h+2
f
h
t
2
+

and therefore:
f(t)
α
= f
α
h
t
αh

1 +
f
h+1
f
h
t +
f
h+2
f
h
t
2
+

α
.
On the contrary, when α / ∈ N, f(t)
α
only can be
performed if f(t) ∈ T
0
. In that case we have:
f(t)
α
=

f
0
+f
1
t +f
2
t
2
+f
3
t
3
+

α
=
= f
α
0

1 +
f
1
f
0
t +
f
2
f
0
t
2
+
f
3
f
0
t
3
+

α
;
note that in this case if f
0
= 1, usually f
α
0
is not ratio-
nal. In any case, we are always reduced to compute
(1 +g(t))
α
and since:
(1 +g(t))
α
=
∞
¸
k=0

α
k

g(t)
k
(3.12.1)
if the coefficients in g(t) are rational numbers, also
the coefficients in (1 + g(t))
α
are, provided α ∈ Q.
These considerations are to be remembered if the op-
eration is realized in some special environment, as
described in the previous section.
Whatever α ∈ R is, the exponents involved in
the right hand member of (3.12.1) are all positive
integer numbers. Therefore, powers can be real-
ized as successive multiplications or Cauchy products.
This gives a straight-forward method for perform-
ing (1 + g(t))
α
, but it is easily seen to take a time
in the order of O(r
3
), since it requires r products
each executed in time O(r
2
). Fortunately, however,
3.13. LOGARITHM AND EXPONENTIAL 37
+
+
+
+
∗
∗
∗
∗



d
d
d



d
d
d



.
.
.
.
.



d
d
d
`

a
0
`

t
0
`

a
1
`

t
1
`

a
2
`

t
2
`

a
n
`

t
n
`

O(t
n
)












d
d
d
Figure 3.2: The tree for a formal power series
J. C. P. Miller has devised an algorithm allowing to
perform (1 + g(t))
α
in time O(r
2
). In fact, let us
write h(t) = a(t)
α
, where a(t) is any formal power
series with a
0
= 1. By differentiating, we obtain
h
′
(t) = αa(t)
α−1
a
′
(t), or, by multiplying everything
by a(t), a(t)h
′
(t) = αh(t)a
′
(t). Therefore, by extract-
ing the coefficient of t
n−1
we find:
n−1
¸
k=0
a
k
(n −k)h
n−k
= α
n−1
¸
k=0
(k + 1)a
k+1
h
n−k−1
We now isolate the term with k = 0 in the left hand
member and the term having k = n − 1 in the right
hand member (a
0
= 1 by hypothesis):
nh
n
+
n−1
¸
k=1
a
k
(n −k)h
n−k
= αna
n
+
n−1
¸
k=1
αka
k
h
n−k
(in the last sum we performed the change of vari-
able k → k −1, in order to have the same indices as
in the left hand member). We now have an expres-
sion for h
n
only depending on (a
1
, a
2
, . . . , a
n
) and
(h
1
, h
2
, . . . , h
n−1
):
h
n
= αa
n
+
1
n
n−1
¸
k=1
((α + 1)k −n) a
k
h
n−k
=
= αa
n
+
n−1
¸
k=1

(α + 1)k
n
−1

a
k
h
n−k
The computation is now straight-forward. We be-
gin by setting h
0
= 1, and then we successively com-
pute h
1
, h
2
, . . . , h
r
(r = n, if n is the number of terms
in (a
1
, a
2
, . . . , a
n
)). The evaluation of h
k
requires a
number of operations in the order O(k), and therefore
the whole procedure works in time O(r
2
), as desired.
The inverse of a series, i.e., (1+g(t))
−1
, is obtained
by setting α = −1. It is worth noting that the previ-
ous formula becomes:
h
k
= −a
k
−
k−1
¸
j=1
a
j
h
k−j
= −
k
¸
j=1
a
j
h
k−j
(h
0
= 1)
and can be used to prove properties of the inverse of
a power series. As a simple example, the reader can
show that the coefficients in (1 −t)
−1
are all 1.
3.13 Logarithm and exponen-
tial
The idea of Miller can be applied to other operations
on formal power series. In the present section we
wish to use it to perform the (natural) logarithm and
the exponentiation of a series. Let us begin with the
logarithm and try to compute ln(1 + g(t)). As we
know, there is a direct way to perform this operation,
i.e.:
ln(1 +g(t)) =
∞
¸
k=1
1
k
g(t)
k
38 CHAPTER 3. FORMAL POWER SERIES
and this formula only requires a series of successive
products. As for the operation of rising to a power,
the procedure needs a time in the order of O(r
3
),
and it is worth considering an alternative approach.
In fact, if we set h(t) = ln(1+g(t)), by differentiating
we obtain h
′
(t) = g
′
(t)/(1 + g(t)), or h
′
(t) = g
′
(t) −
h
′
(t)g(t). We can now extract the coefficient of t
k−1
and obtain:
kh
k
= kg
k
−
k−1
¸
j=0
(k −j)h
k−j
g
j
However, g
0
= 0 by hypothesis, and therefore we have
an expression relating h
k
to (g
1
, g
2
, . . . , g
k
) and to
(h
1
, h
2
, . . . , h
k−1
):
h
k
= g
k
−
1
k
k−1
¸
j=1
(k −j)h
k−j
g
j
=
= g
k
−
k−1
¸
j=1

1 −
j
k

h
k−j
g
j
A program to perform the logarithm of a formal
power series 1+g(t) begins by setting h
0
= 0 and then
proceeds computing h
1
, h
2
, . . . , h
r
if r is the number
of significant terms in g(t). The total time is clearly
in the order of O(r
2
).
A similar technique can be applied to the com-
putation of exp(g(t)) provided that g(t) / ∈ T
0
. If
g(t) ∈ T
0
, i.e., g(t) = g
0
+ g
1
t + g
2
t
2
+ , we have
exp(g
0
+g
1
t+g
2
t
2
+ ) = e
g0
exp(g
1
t+g
2
t
2
+ ). In
this way we are reduced to the previous case, but we
no longer have rational coefficients when g(t) ∈ Q[[t]].
By differentiating the identity h(t) = exp(g(t)) we
obtain h
′
(t) = g
′
(t) exp(g(t)) = g
′
(t)h(t). We extract
the coefficient of t
k−1
:
kh
k
=
k−1
¸
j=0
(j + 1)g
j+1
h
k−j−1
=
k
¸
j=1
jg
j
h
k−j
This formula allows us to compute h
k
in terms of
(g
1
, g
2
, . . . , g
k
) and (h
0
= 1, h
1
, h
2
, . . . , h
k−1
). A pro-
gram performing exponentiation can be easily writ-
ten by defining h
0
= 1 and successively evaluating
h
1
, h
2
, . . . , h
r
. if r is the number of significant terms
in g(t). Time complexity is obviously O(r
2
).
Unfortunately, a similar trick does not work for
series composition. To compute f(g(t)), when g(t) / ∈
T
0
, we have to resort to the defining formula:
f(g(t)) =
∞
¸
k=0
f
k
g(t)
k
This requires the successive computation of the in-
teger powers of g(t), which can be performed by re-
peated applications of the Cauchy product. The exe-
cution time is in the order of O(r
3
), if r is the minimal
number of significant terms in f(t) and g(t), respec-
tively.
We conclude this section by sketching the obvious
algorithms to compute differentiation and integration
of a formal power series f(t) = f
0
+f
1
t+f
2
t
2
+f
3
t
3
+
. If h(t) = f
′
(t), we have:
h
k
= (k + 1)f
k+1
and therefore the number of significant terms is re-
duced by 1. Conversely, if h(t) =

t
0
f(τ)dτ, we have:
h
k
=
1
k
f
k−1
and h
0
= 0; consequently the number of significant
terms is increased by 1.
Chapter 4
Generating Functions
4.1 General Rules
Let us consider a sequence of numbers F =
(f
0
, f
1
, f
2
, . . .) = (f
k
)
k∈N
; the (ordinary) generat-
ing function for the sequence F is defined as f(t) =
f
0
+ f
1
t + f
2
t
2
+ , where the indeterminate t is
arbitrary. Given the sequence (f
k
)
k∈N
, we intro-
duce the generating function operator (, which ap-
plied to (f
k
)
k∈N
produces the ordinary generating
function for the sequence, i.e., ((f
k
)
k∈N
= f(t). In
this expression t is a bound variable, and a more ac-
curate notation would be (
t
(f
k
)
k∈N
= f(t). This
notation is essential when (f
k
)
k∈N
depends on some
parameter or when we consider multivariate gener-
ating functions. In this latter case, for example, we
should write (
t,w
(f
n,k
)
n,k∈N
= f(t, w) to indicate the
fact that f
n,k
in the double sequence becomes the co-
efficient of t
n
w
k
in the function f(t, w). However,
whenever no ambiguity can arise, we will use the no-
tation ((f
k
) = f(t), understanding also the binding
for the variable k. For the sake of completeness, we
also define the exponential generating function of the
sequence (f
0
, f
1
, f
2
, . . .) as:
c(f
k
) = (

f
k
k!

=
∞
¸
k=0
f
k
t
k
k!
.
The operator ( is clearly linear. The function f(t)
can be shifted or differentiated. Two functions f(t)
and g(t) can be multiplied and composed. This leads
to the properties for the operator (listed in Table 4.1.
Note that formula (G5) requires g
0
= 0. The first
five formulas are easily verified by using the intended
interpretation of the operator (; the last formula can
be proved by means of the LIF, in the form relative
to the composition F(w(t)). In fact we have:
[t
n
]F(t)φ(t)
n
= [t
n−1
]
F(t)
t
φ(t)
n
=
= n[t
n
]
¸
F(y)
y
dy


y = w(t)

;
in the last passage we applied backwards the formula:
[t
n
]F(w(t)) =
1
n
[t
n−1
]F
′
(t)φ(t)
n
(w = tφ(w))
and therefore w = w(t) ∈ T
1
is the unique solution
of the functional equation w = tφ(w). By now apply-
ing the rule of differentiation for the “coefficient of”
operator, we can go on:
[t
n
]F(t)φ(t)
n
= [t
n−1
]
d
dt
¸
F(y)
y
dy


y = w(t)

=
= [t
n−1
]
¸
F(w)
w


w = tφ(w)

dw
dt
.
We have applied the chain rule for differentiation, and
from w = tφ(w) we have:
dw
dt
= φ(w) +t
¸
dφ
dw


w = tφ(w)

dw
dt
.
We can therefore compute the derivative of w(t):
dw
dt
=
¸
φ(w)
1 −tφ
′
(w)


w = tφ(w)

where φ
′
(w) denotes the derivative of φ(w) with re-
spect to w. We can substitute this expression in the
formula above and observe that w/φ(w) = t can be
taken outside of the substitution symbol:
[t
n
]F(t)φ(t)
n
=
= [t
n−1
]
¸
F(w)
w
φ(w)
1 −tφ
′
(w)


w = tφ(w)

=
= [t
n−1
]
1
t
¸
F(w)
1 −tφ
′
(w)


w = tφ(w)

=
= [t
n
]
¸
F(w)
1 −tφ
′
(w)


w = tφ(w)

which is our diagonalization rule (G6).
The name “diagonalization” is due to the fact
that if we imagine the coefficients of F(t)φ(t)
n
as
constituting the row n in an infinite matrix, then
39
40 CHAPTER 4. GENERATING FUNCTIONS
linearity ((αf
k
+βg
k
) = α((f
k
) +β((g
k
) (G1)
shifting ((f
k+1
) =
(((f
k
) −f
0
)
t
(G2)
differentiation ((kf
k
) = tD((f
k
) (G3)
convolution (

n
¸
k=0
f
k
g
n−k

= ((f
k
) ((g
k
) (G4)
composition
∞
¸
n=0
f
n
(((g
k
))
n
= ((f
k
) ◦ ((g
k
) (G5)
diagonalisation (([t
n
]F(t)φ(t)
n
) =
¸
F(w)
1 −tφ
′
(w)


w = tφ(w)

(G6)
Table 4.1: The rules for the generating function operator
[t
n
]F(t)φ(t)
n
are just the elements in the main di-
agonal of this array.
The rules (G1) −(G6) can also be assumed as ax-
ioms of a theory of generating functions and used to
derive general theorems as well as specific functions
for particular sequences. In the next sections, we
will prove a number of properties of the generating
function operator. The proofs rely on the following
fundamental principle of identity:
Given two sequences (f
k
)
k∈N
and (g
k
)
k∈N
, then
((f
k
) = ((g
k
) if and only if for every k ∈ N f
k
= g
k
.
The principle is rather obvious from the very defini-
tion of the concept of generating functions; however,
it is important, because it states the condition under
which we can pass from an identity about elements
to the corresponding identity about generating func-
tions. It is sufficient that the two sequences do not
agree by a single element (e.g., the first one) and we
cannot infer the equality of generating functions.
4.2 Some Theorems on Gener-
ating Functions
We are now going to prove a series of properties of
generating functions.
Theorem 4.2.1 Let f(t) = ((f
k
) be the generating
function of the sequence (f
k
)
k∈N
, then
((f
k+2
) =
((f
k
) −f
0
−f
1
t
t
2
(4.2.1)
Proof: Let g
k
= f
k+1
; by (G2), ((g
k
) =
(((f
k
) −f
0
) /t. Since g
0
= f
1
, we have:
((f
k+2
) = ((g
k+1
) =
((g
k
) −g
0
t
=
((f
k
) −f
0
−f
1
t
t
2
By mathematical induction this result can be gen-
eralized to:
Theorem 4.2.2 Let f(t) = ((f
k
) be as above; then:
((f
k+j
) =
((f
k
) −f
0
−f
1
t − −f
j−1
t
j−1
t
j
(4.2.2)
If we consider right instead of left shifting we have
to be more careful:
Theorem 4.2.3 Let f(t) = ((f
k
) be as above, then:
((f
k−j
) = t
j
((f
k
) (4.2.3)
Proof: We have ((f
k
) = (

f
(k−1)+1

=
t
−1
(((f
n−1
) −f
−1
) where f
−1
is the coefficient of t
−1
in f(t). If f(t) ∈ T, f
−1
= 0 and ((f
k−1
) = t((f
k
).
The theorem then follows by mathematical induction.
Property (G3) can be generalized in several ways:
Theorem 4.2.4 Let f(t) = ((f
k
) be as above; then:
(((k + 1)f
k+1
) = D((f
k
) (4.2.4)
Proof: If we set g
k
= kf
k
, we obtain:
(((k + 1)f
k+1
) = ((g
k+1
) =
= t
−1
(((kf
k
) −0f
0
)
(G2)
=
= t
−1
tD((f
k
) = D((f
k
) .
Theorem 4.2.5 Let f(t) = ((f
k
) be as above; then:
(

k
2
f
k

= tD((f
k
) +t
2
D
2
((f
k
) (4.2.5)
This can be further generalized:
4.3. MORE ADVANCED RESULTS 41
Theorem 4.2.6 Let f(t) = ((f
k
) be as above; then:
(

k
j
f
k

= o
j
(tD)((f
k
) (4.2.6)
where o
j
(w) =
¸
j
r=1
¸
j
r
¸
w
r
is the jth Stirling poly-
nomial of the second kind (see Section 2.11).
Proof: Formula (4.2.6) is to be understood in the
operator sense; so, for example, being o
3
(w) = w +
3w
2
+w
3
, we have:
(

k
3
f
k

= tD((f
k
) + 3t
2
D
2
((f
k
) +t
3
D
3
((f
k
) .
The proof proceeds by induction, as (G3) and (4.2.5)
are the first two instances. Now:
(

k
j+1
f
k

= (

k(k
j
)f
k

= tD(

k
j
f
k

that is:
o
j+1
(tD) = tDo
j
(tD) = tD
j
¸
r=1
S(j, r)t
r
D
r
=
=
j
¸
r=1
S(j, r)rt
r
D
r
+
j
¸
r=1
S(j, r)t
r+1
D
r+1
.
By equating like coefficients we find S(j + 1, r) =
rS(j, r) + S(j, r − 1), which is the classical recur-
rence for the Stirling numbers of the second kind.
Since initial conditions also coincide, we can conclude
S(j, r) =
¸
j
r
¸
.
For the falling factorial k
r
= k(k − 1) (k − r +
1) we have a simpler formula, the proof of which is
immediate:
Theorem 4.2.7 Let f(t) = ((f
k
) be as above; then:
((k
r
f
k
) = t
r
D
r
((f
k
) (4.2.7)
Let us now come to integration:
Theorem 4.2.8 Let f(t) = ((f
k
) be as above and
let us define g
k
= f
k
/k, ∀k = 0 and g
0
= 0; then:
(

1
k
f
k

= ((g
k
) =

t
0
(((f
k
) −f
0
)
dz
z
(4.2.8)
Proof: Clearly, kg
k
= f
k
, except for k = 0. Hence
we have ((kg
k
) = ((f
k
) −f
0
. By using (G3), we find
tD((g
k
) = ((f
k
) −f
0
, from which (4.2.8) follows by
integration and the condition g
0
= 0.
A more classical formula is:
Theorem 4.2.9 Let f(t) = ((f
k
) be as above or,
equivalently, let f(t) be a f.p.s. but not a f.L.s.; then:
(

1
k + 1
f
k

=
=
1
t

t
0
((f
k
) dz =
1
t

t
0
f(z) dz (4.2.9)
Proof: Let us consider the sequence (g
k
)
k∈N
, where
g
k+1
= f
k
and g
0
= 0. So we have: ((g
k+1
) =
((f
k
) = t
−1
(((g
k
) −g
0
). Finally:
(

1
k + 1
f
k

= (

1
k + 1
g
k+1

=
1
t
(

1
k
g
k

=
=
1
t

t
0
(((g
k
) −g
0
)
dz
z
=
1
t

t
0
((f
k
) dz
In the following theorems, f(t) will always denote
the generating function ((f
k
).
Theorem 4.2.10 Let f(t) = ((f
k
) denote the gen-
erating function of the sequence (f
k
)
k∈N
; then:
(

p
k
f
k

= f(pt) (4.2.10)
Proof: By setting g(t) = pt in (G5) we have:
f(pt) =
¸
∞
n=0
f
n
(pt)
n
=
¸
∞
n=0
p
n
f
n
t
n
= (

p
k
f
k

In particular, for p = −1 we have: (

(−1)
k
f
k

=
f(−t).
4.3 More advanced results
The results obtained till now can be considered as
simple generalizations of the axioms. They are very
useful and will be used in many circumstances. How-
ever, we can also obtain more advanced results, con-
cerning sequences derived from a given sequence by
manipulating its elements in various ways. For exam-
ple, let us begin by proving the well-known bisection
formulas:
Theorem 4.3.1 Let f(t) = ((f
k
) denote the gener-
ating function of the sequence (f
k
)
k∈N
; then:
((f
2k
) =
f(
√
t) +f(−
√
t)
2
(4.3.1)
((f
2k+1
) =
f(
√
t) −f(−
√
t)
2
√
t
(4.3.2)
where
√
t is a symbol with the property that (
√
t)
2
= t.
Proof: By (G5) we have:
f(
√
t) +f(−
√
t)
2
=
=
¸
∞
n=0
f
n
√
t
n
+
¸
∞
n=0
f
n
(−
√
t)
n
2
=
=
∞
¸
n=0
f
n
(
√
t)
n
+ (−
√
t)
n
2
For n odd (
√
t)
n
+ (−
√
t)
n
= 0; hence, by setting
n = 2k, we have:
∞
¸
k=0
f
2k
(t
k
+t
k
)/2 =
∞
¸
k=0
f
2k
t
k
= ((f
2k
) .
42 CHAPTER 4. GENERATING FUNCTIONS
The proof of the second formula is analogous.
The following proof is typical and introduces the
use of ordinary differential equations in the calculus
of generating functions:
Theorem 4.3.2 Let f(t) = ((f
k
) denote the gener-
ating function of the sequence (f
k
)
k∈N
; then:
(

f
k
2k + 1

=
1
2
√
t

t
0
((f
k
)
√
z
dz (4.3.3)
Proof: Let us set g
k
= f
k
/(2k+1), or 2kg
k
+g
k
= f
k
.
If g(t) = ((g
k
), by applying (G3) we have the differ-
ential equation 2tg
′
(t) + g(t) = f(t), whose solution
having g(0) = f
0
is just formula (4.3.3).
We conclude with two general theorems on sums:
Theorem 4.3.3 (Partial Sum Theorem) Let
f(t) = ((f
k
) denote the generating function of the
sequence (f
k
)
k∈N
; then:
(

n
¸
k=0
f
k

=
1
1 −t
((f
k
) (4.3.4)
Proof: If we set s
n
=
¸
n
k=0
f
k
, then we have s
n+1
=
s
n
+f
n+1
for every n ∈ N and we can apply the oper-
ator ( to both members: ((s
n+1
) = ((s
n
) +((f
n+1
),
i.e.:
((s
n
) −s
0
t
= ((s
n
) +
((f
n
) −f
0
t
Since s
0
= f
0
, we find ((s
n
) = t((s
n
) + ((f
n
) and
from this (4.3.4) follows directly.
The following result is known as Euler transforma-
tion:
Theorem 4.3.4 Let f(t) = ((f
k
) denote the gener-
ating function of the sequence (f
k
)
k∈N
; then:
(

n
¸
k=0

n
k

f
k

=
1
1 −t
f

t
1 −t

(4.3.5)
Proof: By well-known properties of binomial coeffi-
cients we have:

n
k

=

n
n −k

=

−n +n −k −1
n −k

(−1)
n−k
=
=

−k −1
n −k

(−1)
n−k
and this is the coefficient of t
n−k
in (1 −t)
−k−1
. We
now observe that the sum in (4.3.5) can be extended
to infinity, and by (G5) we have:
n
¸
k=0

n
k

f
k
=
n
¸
k=0

−k −1
n −k

(−1)
n−k
f
k
=
=
∞
¸
k=0
[t
n−k
](1 −t)
−k−1
[y
k
]f(y) =
= [t
n
]
1
1 −t
∞
¸
k=0
[y
k
]f(y)

t
1 −t

k
=
= [t
n
]
1
1 −t
f

t
1 −t

.
Since the last expression does not depend on n, it
represents the generating function of the sum.
We observe explicitly that by (K4) we have:
¸
n
k=0

n
k

f
k
= [t
n
](1 + t)
n
f(t), but this expression
does not represent a generating function because it
depends on n. The Euler transformation can be gen-
eralized in several ways, as we shall see when dealing
with Riordan arrays.
4.4 Common Generating Func-
tions
The aim of the present section is to derive the most
common generating functions by using the apparatus
of the previous sections. As a first example, let us
consider the constant sequence F = (1, 1, 1, . . .), for
which we have f
k+1
= f
k
for every k ∈ N. By ap-
plying the principle of identity, we find: ((f
k+1
) =
((f
k
), that is by (G2): ((f
k
) − f
0
= t((f
k
). Since
f
0
= 1, we have immediately:
((1) =
1
1 −t
For any constant sequence F = (c, c, c, . . .), by (G1)
we find that ((c) = c(1−t)
−1
. Similarly, by using the
basic rules and the theorems of the previous sections
we have:
((n) = ((n 1) = tD
1
1 −t
=
t
(1 −t)
2
(

n
2

= tD((n) = tD
1
(1 −t)
2
=
t +t
2
(1 −t)
3
(((−1)
n
) = ((1) ◦ (−t) =
1
1 +t
(

1
n

= (

1
n
1

=

t
0

1
1 −z
−1

dz
z
=
=

t
0
dz
1 −z
= ln
1
1 −t
((H
n
) = (

n
¸
k=0
1
k

=
1
1 −t
(

1
n

=
=
1
1 −t
ln
1
1 −t
where H
n
is the nth harmonic number. Other gen-
erating functions can be obtained from the previous
formulas:
((nH
n
) = tD
1
1 −t
ln
1
1 −t
=
4.4. COMMON GENERATING FUNCTIONS 43
=
t
(1 −t)
2

ln
1
1 −t
+ 1

(

1
n + 1
H
n

=
1
t

t
0
1
1 −z
ln
1
1 −z
dz =
=
1
2t

ln
1
1 −t

2
((δ
0,n
) = ((1) −t((1) =
1 −t
1 −t
= 1
where δ
n,m
is the Kronecker’s delta. This last re-
lation can be readily generalized to ((δ
n,m
) = t
m
.
An interesting example is given by (

1
n(n+1)

. Since
1
n(n+1)
=
1
n
−
1
n+1
, it is tempting to apply the op-
erator ( to both members. However, this relation is
not valid for n = 0. In order to apply the principle
of identity, we must define:
1
n(n + 1)
=
1
n
−
1
n + 1
+δ
n,0
in accordance with the fact that the first element of
the sequence is zero. We thus arrive to the correct
generating function:
(

1
n(n + 1)

= 1 −
1 −t
t
ln
1
1 −t
Let us now come to binomial coefficients. In order
to find (

p
k

we observe that, from the definition:

p
k + 1

=
p(p −1) (p −k + 1)(p −k)
(k + 1)!
=
=
p −k
k + 1

p
k

.
Hence, by denoting

p
k

as f
k
, we have:
(((k + 1)f
k+1
) = (((p −k)f
k
) = p((f
k
) − ((kf
k
).
By applying (4.2.4) and (G3) we have
D((f
k
) = p((f
k
) − tD((f
k
), i.e., the differen-
tial equation f
′
(t) = pf(t) − tf
′
(t). By sep-
arating the variables and integrating, we find
ln f(t) = p ln(1 +t) +c, or f(t) = c(1 +t)
p
. For t = 0
we should have f(0) =

p
0

= 1, and this implies
c = 1. Consequently:
(

p
k

= (1 +t)
p
p ∈ R
We are now in a position to derive the recurrence
relation for binomial coefficients. By using (K1) ÷
(K5) we find easily:

p
k

= [t
k
](1 +t)
p
= [t
k
](1 +t)(1 +t)
p−1
=
= [t
k
](1 +t)
p−1
+ [t
k−1
](1 +t)
p−1
=
=

p −1
k

+

p −1
k −1

.
By (K4), we have the well-known Vandermonde
convolution:

m+p
n

= [t
n
](1 +t)
m+p
=
= [t
n
](1 +t)
m
(1 +t)
p
=
=
n
¸
k=0

m
k

p
n −k

which, for m = p = n becomes
¸
n
k=0

n
k

2
=

2n
n

.
We can also find (


k
p


, where p is a parameter.
The derivation is purely algebraic and makes use of
the generating functions already found and of various
properties considered in the previous section:
(

k
p

= (

k
k −p

=
= (

−k +k −p + 1
k −p

(−1)
k−p

=
= (

−p −1
k −p

(−1)
k−p

=
= (

[t
k−p
]
1
(1 −t)
p+1

=
= (

[t
k
]
t
p
(1 −t)
p+1

=
t
p
(1 −t)
p+1
.
Several generating functions for different forms of bi-
nomial coefficients can be found by means of this
method. They are summarized as follows, where p
and m are two parameters and can also be zero:
(

p
m+k

=
(1 +t)
p
t
m
(

p +k
m

=
t
m−p
(1 −t)
m+1
(

p +k
m+k

=
1
t
m
(1 −t)
p+1−m
These functions can make sense even when they are
f.L.s. and not simply f.p.s..
Finally, we list the following generating functions:
(

k

p
k

= tD(

p
k

=
= tD(1 +t)
p
= pt(1 +t)
p−1
(

k
2

p
k

= (tD +t
2
D
2
)(

p
k

=
= pt(1 +pt)(1 +t)
p−2
(

1
k + 1

p
k

=
1
t

t
0
(

p
k

dz =
=
1
t
¸
(1 +t)
p+1
p + 1

t
0
=
=
(1 +t)
p+1
−1
(p + 1)t
44 CHAPTER 4. GENERATING FUNCTIONS
(

k

k
m

= tD
t
m
(1 −t)
m+1
=
=
mt
m
+t
m+1
(1 −t)
m+2
(

k
2

k
m

= (tD +t
2
D
2
)(

k
m

=
=
m
2
t
m
+ (3m+ 1)t
m+1
+t
m+2
(1 −t)
m+3
(

1
k

k
m

=

t
0

(

k
m

−

0
m

dz
z
=
=

t
0
z
m−1
dz
(1 −z)
m+1
=
t
m
m(1 −t)
m
The last integral can be solved by setting y = (1 −
z)
−1
and is valid for m > 0; for m = 0 it reduces to
((1/k) = −ln(1 −t).
4.5 The Method of Shifting
When the elements of a sequence F are given by an
explicit formula, we can try to find the generating
function for F by using the technique of shifting: we
consider the element f
n+1
and try to express it in
terms of f
n
. This can produce a relation to which
we apply the principle of identity deriving an equa-
tion in ((f
n
), the solution of which is the generating
function. In practice, we find a recurrence for the
elements f
n
∈ F and then try to solve it by using
the rules (G1) ÷(G5) and their consequences. It can
happen that the recurrence involves several elements
in F and/or that the resulting equation is indeed a
differential equation. Whatever the case, the method
of shifting allows us to find the generating function
of many sequences.
Let us consider the geometric sequence

1, p, p
2
, p
3
, . . .

; we have p
k+1
= pp
k
, ∀k ∈ N
or, by applying the operator (, (

p
k+1

= p(

p
k

.
By (G2) we have t
−1

(

p
k

−1

= p(

p
k

, that is:
(

p
k

=
1
1 −pt
From this we obtain other generating functions:
(

kp
k

= tD
1
1 −pt
=
pt
(1 −pt)
2
(

k
2
p
k

= tD
pt
(1 −pt)
2
=
pt +p
2
t
2
(1 −pt)
3
(

1
k
p
k

=

t
0

1
1 −pz
−1

dz
z
=
=

p dz
(1 −pz)
= ln
1
1 −pt
(

n
¸
k=0
p
k

=
1
1 −t
1
1 −pt
=
=
1
(p −1)t

1
1 −pt
−
1
1 −t

.
The last relation has been obtained by partial fraction
expansion. By using the operator [t
k
] we easily find:
n
¸
k=0
p
k
= [t
n
]
1
1 −t
1
1 −pt
=
=
1
(p −1)
[t
n+1
]

1
1 −pt
−
1
1 −t

=
=
p
n+1
−1
p −1
the well-known formula for the sum of a geometric
progression. We observe explicitly that the formulas
above could have been obtained from formulas of the
previous section and the general formula (4.2.10). In
a similar way we also have:
(

m
k

p
k

= (1 +pt)
m
(

k
m

p
k

= (

k
k −m

p
k−m
p
m

=
= p
m
(

−m−1
k −m

(−p)
k−m

=
=
p
m
t
m
(1 −pt)
m+1
.
As a very simple application of the shifting method,
let us observe that:
1
(n + 1)!
=
1
n + 1
1
n!
that is (n+1)f
n+1
= f
n
, where f
n
= 1/n!. By (4.2.4)
we have f
′
(t) = f(t) or:
(

1
n!

= e
t
(

1
n n!

=

t
0
e
z
−1
z
dz
(

n
(n + 1)!

= tD(

1
(n + 1)!

=
= tD
1
t
(e
t
−1) =
te
t
−e
t
+ 1
t
By this relation the well-known result follows:
n
¸
k=0
k
(k + 1)!
= [t
n
]
1
1 −t

te
t
−e
t
+ 1
t

=
= [t
n
]

1
1 −t
−
e
t
−1
t

=
= 1 −
1
(n + 1)!
.
Let us now observe that

2n+2
n+1

=
2(2n+1)
n+1

2n
n

;
by setting f
n
=

2n
n

, we have the recurrence (n +
4.6. DIAGONALIZATION 45
1)f
n+1
= 2(2n + 1)f
n
. By using (4.2.4), (G1) and
(G3) we obtain the differential equation: f
′
(t) =
4tf
′
(t) + 2f(t), the simple solution of which is:
(

2n
n

=
1
√
1 −4t
(

1
n + 1

2n
n

=
1
t

t
0
dz
√
1 −4z
=
1 −
√
1 −4t
2t
(

1
n

2n
n

=

t
0

1
√
1 −4z
−1

dz
z
=
= 2 ln
1 −
√
1 −4t
2t
(

n

2n
n

=
2t
(1 −4t)
√
1 −4t
(

1
2n + 1

2n
n

=
1
√
t

t
0
dz

4z(1 −4z)
=
=
1
√
4t
arctan

4t
1 −4t
.
A last group of generating functions is obtained by
considering f
n
= 4
n

2n
n

−1
. Since:
4
n+1

2n + 2
n + 1

−1
=
2n + 2
2n + 1
4
n

2n
n

−1
we have the recurrence: (2n + 1)f
n+1
= 2(n + 1)f
n
.
By using the operator ( and the rules of Section 4.2,
the differential equation 2t(1−t)f
′
(t) −(1+2t)f(t) +
1 = 0 is derived. The solution is:
f(t) =

t
(1 −t)
3

−

t
0

(1 −z)
3
z
dz
2z(1 −z)

By simplifying and using the change of variable y =

z/(1 −z), the integral can be computed without
difficulty, and the final result is:
(

4
n

2n
n

−1

=

t
(1 −t)
3
arctan

t
1 −t
+
1
1 −t
.
Some immediate consequences are:
(

4
n
2n + 1

2n
n

−1

=
1

t(1 −t)
arctan

t
1 −t
(

4
n
2n
2

2n
n

−1

=

arctan

t
1 −t

2
and finally:
(

1
2n
4
n
2n + 1

2n
n

−1

= 1−

1 −t
t
arctan

t
1 −t
.
4.6 Diagonalization
The technique of shifting is a rather general method
for obtaining generating functions. It produces first
order recurrence relations, which will be more closely
studied in the next sections. Not every sequence can
be defined by a first order recurrence relation, and
other methods are often necessary to find out gener-
ating functions. Sometimes, the rule of diagonaliza-
tion can be used very conveniently. One of the most
simple examples is how to determine the generating
function of the central binomial coefficients, without
having to pass through the solution of a differential
equation. In fact we have:

2n
n

= [t
n
](1 +t)
2n
and (G6) can be applied with F(t) = 1 and φ(t) =
(1 +t)
2
. In this case, the function w = w(t) is easily
determined by solving the functional equation w =
t(1+w)
2
. By expanding, we find tw
2
−(1−2t)w+t = 0
or:
w = w(t) =
1 −t ±
√
1 −4t
2t
.
Since w = w(t) should belong to T
1
, we must elim-
inate the solution with the + sign; consequently, we
have:
(

2n
n

=
=
¸
1
1 −2t(1 +w)


w =
1 −t −
√
1 −4t
2t

=
=
1
√
1 −4t
as we already know.
The function φ(t) = (1 +t)
2
gives rise to a second
degree equation. More in general, let us study the
sequence:
c
n
= [t
n
](1 +αt +βt
2
)
n
and look for its generating function C(t) = ((c
n
).
In this case again we have F(t) = 1 and φ(t) = 1 +
αt +βt
2
, and therefore we should solve the functional
equation w = t(1+αw+βw
2
) or βtw
2
−(1−αt)w+t =
0. This gives:
w = w(t) =
1 −αt ±

(1 −αt)
2
−4βt
2
2βt
and again we have to eliminate the solution with the
+ sign. By performing the necessary computations,
we find:
C(t) =
¸
1
1 −t(α + 2βw)



46 CHAPTER 4. GENERATING FUNCTIONS
n`k 0 1 2 3 4 5 6 7 8
0 1
1 1 1 1
2 1 2 3 2 1
3 1 3 6 7 6 3 1
4 1 4 10 16 19 16 10 4 1
Table 4.2: Trinomial coefficients


w =
1 −αt −

1 −2αt + (α
2
−4β)t
2
2βt
¸
=
1

1 −2αt + (α
2
−4β)t
2
and for α = 2, β = 1 we obtain again the generating
function for the central binomial coefficients.
The coefficients of (1+t +t
2
)
n
are called trinomial
coefficients, in analogy with the binomial coefficients.
They constitute an infinite array in which every row
has two more elements, different from 0, with respect
to the previous row (see Table 4.2).
If T
n,k
is [t
k
](1 + t + t
2
)
n
, a trinomial coefficient,
by the obvious property (1 + t + t
2
)
n+1
= (1 + t +
t
2
)(1+t+t
2
)
n
, we immediately deduce the recurrence
relation:
T
n+1,k+1
= T
n,k−1
+T
n,k
+T
n,k+1
from which the array can be built, once we start from
the initial conditions T
n,0
= 1 and T
n,2n
= 1, for ev-
ery n ∈ N. The elements T
n,n
, marked in the table,
are called the central trinomial coefficients; their se-
quence begins:
n 0 1 2 3 4 5 6 7 8
T
n
1 1 3 7 19 51 141 393 1107
and by the formula above their generating function
is:
((T
n,n
) =
1
√
1 −2t −3t
2
=
1

(1 +t)(1 −3t)
.
4.7 Some special generating
functions
We wish to determine the generating function of
the sequence ¦0, 1, 1, 1, 2, 2, 2, 2, 3, . . .¦, that is the
sequence whose generic element is ⌊
√
k⌋. We can
think that it is formed up by summing an infinite
number of simpler sequences ¦0, 1, 1, 1, 1, 1, 1, . . .¦,
¦0, 0, 0, 0, 1, 1, 1, 1, . . .¦, the next one with the first 1
in position 9, and so on. The generating functions of
these sequences are:
t
1
1 −t
t
4
1 −t
t
9
1 −t
t
16
1 −t

and therefore we obtain:
(

⌊
√
k⌋

=
∞
¸
k=0
t
k
2
1 −t
.
In the same way we obtain analogous generating
functions:
(

⌊
r
√
k⌋

=
∞
¸
k=0
t
k
r
1 −t
((⌊log
r
k⌋) =
∞
¸
k=0
t
r
k
1 −t
where r is any integer number, or also any real num-
ber, if we substitute to k
r
and r
k
, respectively, ⌈k
r
⌉
and ⌈r
k
⌉.
These generating functions can be used to find the
values of several sums in closed or semi-closed form.
Let us begin by the following case, where we use the
Euler transformation:
¸
k

n
k

⌊
√
k⌋(−1)
k
=
= (−1)
n
¸
k

n
k

(−1)
n−k
⌊
√
k⌋ =
= (−1)
n
[t
n
]
1
1 +t
¸
∞
¸
k=0
y
k
2
1 −y


y =
t
1 +t
¸
=
= (−1)
n
[t
n
]
∞
¸
k=0
t
k
2
(1 +t)
k
2
=
= (−1)
n
⌊
√
n⌋
¸
k=0
[t
n−k
2
]
1
(1 +t)
k
2
=
= (−1)
n
⌊
√
n⌋
¸
k=0

−k
2
n −k
2

=
= (−1)
n
⌊
√
n⌋
¸
k=0

n −1
n −k
2

(−1)
n−k
2
=
=
⌊
√
n⌋
¸
k=0

n −1
n −k
2

(−1)
k
2
.
We can think of the last sum as a “semi-closed”
form, because the number of terms is dramatically
reduced from n to
√
n, although it remains depending
on n. In the same way we find:
¸
k

n
k

⌊log
2
k⌋(−1)
k
=
⌊log
2
n⌋
¸
k=1

n −1
n −2
k

.
A truly closed formula is found for the following
sum:
n
¸
k=1
⌊
√
k⌋ = [t
n
]
1
1 −t
∞
¸
k=1
t
k
2
1 −t
=
4.8. LINEAR RECURRENCES WITH CONSTANT COEFFICIENTS 47
=
∞
¸
k=1
[t
n−k
2
]
1
(1 −t)
2
=
=
⌊
√
n⌋
¸
k=1

−2
n −k
2

(−1)
n−k
2
=
=
⌊
√
n⌋
¸
k=1

n −k
2
+ 1
n −k
2

=
=
⌊
√
n⌋
¸
k=1
(n −k
2
+ 1) =
= (n + 1)⌊
√
n⌋ −
⌊
√
n⌋
¸
k=1
k
2
.
The final value of the sum is therefore:
(n + 1)⌊
√
n⌋ −
⌊
√
n⌋
3

⌊
√
n⌋ + 1


⌊
√
n⌋ +
1
2

whose asymptotic value is
2
3
n
√
n. Again, for the anal-
ogous sum with ⌊log
2
n⌋, we obtain (see Chapter 1):
n
¸
k=1
⌊log
2
k⌋ = (n + 1)⌊log
2
n⌋ −2
⌊log
2
n⌋+1
+ 1.
A somewhat more difficult sum is the following one:
n
¸
k=0

n
k

⌊
√
k⌋ = [t
n
]
1
1 −t
¸
∞
¸
k=0
y
k
2
1 −y


y =
t
1 −t
¸
= [t
n
]
∞
¸
k=0
t
k
2
(1 −t)
k
2
(1 −2t)
=
=
∞
¸
k=0
[t
n−k
2
]
1
(1 −t)
k
2
(1 −2t)
.
We can now obtain a semi-closed form for this sum by
expanding the generating function into partial frac-
tions:
1
(1 −t)
k
2
(1 −2t)
=
A
1 −2t
+
B
(1 −t)
k
2
+
+
C
(1 −t)
k
2
−1
+
D
(1 −t)
k
2
−2
+ +
X
1 −t
.
We can show that A = 2
k
2
, B = −1, C = −2, D =
−4, . . . , X = −2
k
2
−1
; in fact, by substituting these
values in the previous expression we get:
2
k
2
1 −2t
−
1
(1 −t)
k
2
−
2(1 −t)
(1 −t)
k
2
−
−
4(1 −t)
2
(1 −t)
k
2
− −
2
k
2
−1
(1 −t)
k
2
−1
(1 −t)
k
2
=
=
2
k
2
1 −2t
−
1
(1 −t)
k
2

2
k
2
(1 −t)
k
2
−1
2(1 −t) −1

=
=
2
k
2
1 −2t
−
2
k
2
(1 −t)
k
2
−1
(1 −t)
k
2
(1 −2t)
=
=
1
(1 −t)
k
2
(1 −2t)
.
Therefore, we conclude:
n
¸
k=0

n
k

⌊
√
k⌋ =
⌊
√
n⌋
¸
k=1
2
k
2
2
n−k
2
−
⌊
√
n⌋
¸
k=1

n −1
n −k
2

−
−
⌊
√
n⌋
¸
k=1
2

n −2
n −k
2

− −
⌊
√
n⌋
¸
k=1
2
k
2
−1

n −k
2
n −k
2

=
= ⌊
√
n⌋2
n
−
⌊
√
n⌋
¸
k=1

n −1
n −k
2

+ 2

n −2
n −k
2

+ +
+ + 2
k
2
−1

n −k
2
n −k
2

.
We observe that for very large values of n the first
term dominates all the others, and therefore the
asymptotic value of the sum is ⌊
√
n⌋2
n
.
4.8 Linear recurrences with
constant coefficients
If (f
k
)
k∈N
is a sequence, it can be defined by means
of a recurrence relation, i.e., a relation relating the
generic element f
n
to other elements f
k
having k < n.
Usually, the first elements of the sequence must be
given explicitly, in order to allow the computation
of the successive values; they constitute the initial
conditions and the sequence is well-defined if and only
if every element can be computed by starting with
the initial conditions and going on with the other
elements by means of the recurrence relation. For
example, the constant sequence (1, 1, 1, . . .) can be
defined by the recurrence relation x
n
= x
n−1
and
the initial condition x
0
= 1. By changing the initial
conditions, the sequence can radically change; if we
consider the same relation x
n
= x
n−1
together with
the initial condition x
0
= 2, we obtain the constant
sequence ¦2, 2, 2, . . .¦.
In general, a recurrence relation can be written
f
n
= F(f
n−1
, f
n−2
, . . .); when F depends on all
the values f
n−1
, f
n−2
, . . . , f
1
, f
0
, then the relation is
called a full history recurrence. If F only depends on
a fixed number of elements f
n−1
, f
n−2
, . . . , f
n−p
, then
the relation is called a partial history recurrence and p
is called the order of the relation. Besides, if F is lin-
ear, we have a linear recurrence. Linear recurrences
are surely the most common and important type of
recurrence relations; if all the coefficients appearing
in F are constant, we have a linear recurrence with
48 CHAPTER 4. GENERATING FUNCTIONS
constant coefficients, and if the coefficients are poly-
nomials in n, we have a linear recurrence with poly-
nomial coefficients. As we are now going to see, the
method of generating functions allows us to find the
solution of any linear recurrence with constant coeffi-
cients, in the sense that we find a function f(t) such
that [t
n
]f(t) = f
n
, ∀n ∈ N. For linear recurrences
with polynomial coefficients, the same method allows
us to find a solution in many occasions, but the suc-
cess is not assured. On the other hand, no method
is known that solves all the recurrences of this kind,
and surely generating functions are the method giv-
ing the highest number of positive results. We will
discuss this case in the next section.
The Fibonacci recurrence F
n
= F
n−1
+F
n−2
is an
example of a recurrence relation with constant co-
efficients. When we have a recurrence of this kind,
we begin by expressing it in such a way that the re-
lation is valid for every n ∈ N. In the example of
Fibonacci numbers, this is not the case, because for
n = 0 we have F
0
= F
−1
+F
−2
, and we do not know
the values for the two elements in the r.h.s., which
have no combinatorial meaning. However, if we write
the recurrence as F
n+2
= F
n+1
+F
n
we have fulfilled
the requirement. This first step has a great impor-
tance, because it allows us to apply the operator (
to both members of the relation; this was not pos-
sible beforehand because of the principle of identity
for generating functions.
The recurrence being linear with constant coeffi-
cients, we can apply the axiom of linearity to the
recurrence:
f
n+p
= α
1
f
n+p−1
+α
2
f
n+p−2
+ +α
p
f
n
and obtain the relation:
((f
n+p
) = α
1
((f
n+p−1
)+
+α
2
((f
n+p−2
) + +α
p
((f
n
).
By Theorem 4.2.2 we can now express every
((f
n+p−j
) in terms of f(t) = ((f
n
) and obtain a lin-
ear relation in f(t), from which an explicit expression
for f(t) is immediately obtained. This is the solution
of the recurrence relation. We observe explicitly that
in writing the expressions for ((f
n+p−j
) we make use
of the initial conditions for the sequence.
Let us go on with the example of the Fibonacci
sequence (F
k
)
k∈N
. We have:
((F
n+2
) = ((F
n+1
) +((F
n
)
and by setting F(t) = ((F
n
) we find:
F(t) −F
0
−F
1
t
t
2
=
F(t) −F
0
t
+F(t).
Because we know that F
0
= 0, F
1
= 1, we have:
F(t) −t = tF(t) +t
2
F(t)
and by solving in F(t) we have the explicit expression:
F(t) =
t
1 −t −t
2
.
This is the generating function for the Fibonacci
numbers. We can now find an explicit expression for
F
n
in the following way. The denominator of F(t)
can be written 1 −t −t
2
= (1 −φt)(1 −
´
φt) where:
φ =
1 +
√
5
2
≈ 1.618033989
´
φ =
1 −
√
5
2
≈ −0.618033989.
The constant 1/φ ≈ 0.618033989 is known as the
golden ratio. By applying the method of partial frac-
tion expansion we find:
F(t) =
t
(1 −φt)(1 −
´
φt)
=
A
1 −φt
+
B
1 −
´
φt
=
=
A−A
´
φt +B −Bφt
1 −t −t
2
.
We determine the two constants A and B by equat-
ing the coefficients in the first and last expression for
F(t):

A+B = 0
−A
´
φ −Bφ = 1

A = 1/(φ −
´
φ) = 1/
√
5
B = −A = −1/
√
5
The value of F
n
is now obtained by extracting the
coefficient of t
n
:
F
n
= [t
n
]F(t) =
= [t
n
]
1
√
5

1
1 −φt
−
1
1 −
´
φt

=
=
1
√
5

[t
n
]
1
1 −φt
−[t
n
]
1
1 −
´
φt

=
=
φ
n
−
´
φ
n
√
5
.
This formula allows us to compute F
n
in a time in-
dependent of n, because φ
n
= exp(nln φ), and shows
that F
n
grows exponentially. In fact, since [
´
φ[ < 1,
the quantity
´
φ
n
approaches 0 very rapidly and we
have F
n
= O(φ
n
). In reality, F
n
should be an inte-
ger and therefore we can compute it by finding the
integer number closest to φ
n
/
√
5; consequently:
F
n
= round

φ
n
√
5

.
4.10. THE SUMMING FACTOR METHOD 49
4.9 Linear recurrences with
polynomial coefficients
When a recurrence relation has polynomial coeffi-
cients, the method of generating functions does not
assure a solution, but no other method is available
to solve those recurrences which cannot be solved by
a generating function approach. Usually, the rule of
differentiation introduces a derivative in the relation
for the generating function and a differential equa-
tion has to be solved. This is the actual problem of
this approach, because the main difficulty just con-
sists in dealing with the differential equation. We
have already seen some examples when we studied
the method of shifting, but here we wish to present
a case arising from an actual combinatorial problem,
and in the next section we will see a very important
example taken from the analysis of algorithms.
When we studied permutations, we introduced the
concept of an involution, i.e., a permutation π ∈ P
n
such that π
2
= (1), and for the number I
n
of involu-
tions in P
n
we found the recurrence relation:
I
n
= I
n−1
+ (n −1)I
n−2
which has polynomial coefficients. The number of
involutions grows very fast and it can be a good idea
to consider the quantity i
n
= I
n
/n!. Therefore, let
us begin by changing the recurrence in such a way
that the principle of identity can be applied, and then
divide everything by (n + 2)!:
I
n+2
= I
n+1
+ (n + 1)I
n
I
n+2
(n + 2)!
=
1
n + 2
I
n+1
(n + 1)!
+
1
n + 2
I
n
n!
.
The recurrence relation for i
n
is:
(n + 2)i
n+2
= i
n+1
+i
n
and we can pass to generating functions.
(((n + 2)i
n+2
) can be seen as the shifting of
(((n + 1)i
n+1
) = i
′
(t), if with i(t) we denote the
generating function of (i
k
)
k∈N
= (I
k
/k!)
k∈N
, which
is therefore the exponential generating function for
(I
k
)
k∈N
. We have:
i
′
(t) −1
t
=
i(t) −1
t
+i(t)
because of the initial conditions i
0
= i
1
= 1, and so:
i
′
(t) = (1 +t)i(t).
This is a simple differential equation with separable
variables and by solving it we find:
ln i(t) = t +
t
2
2
+C or i(t) = exp

t +
t
2
2
+C

where C is an integration constant. Because i(0) =
e
C
= 1, we have C = 0 and we conclude with the
formula:
I
n
= n![t
n
] exp

t +
t
2
2

.
4.10 The summing factor
method
For linear recurrences of the first order a method ex-
ists, which allows us to obtain an explicit expression
for the generic element of the defined sequence. Usu-
ally, this expression is in the form of a sum, and a pos-
sible closed form can only be found by manipulating
this sum; therefore, the method does not guarantee
a closed form. Let us suppose we have a recurrence
relation:
a
n+1
f
n+1
= b
n
f
n
+c
n
where a
n
, b
n
, c
n
are any expressions, possibly depend-
ing on n. As we remarked in the Introduction, if
a
n+1
= b
n
= 1, by unfolding the recurrence we can
find an explicit expression for f
n+1
or f
n
:
f
n+1
= f
n
+c
n
= f
n−1
+c
n−1
+c
n
= = f
0
+
n
¸
k=0
c
k
where f
0
is the initial condition relative to the se-
quence under consideration. Fortunately, we can al-
ways change the original recurrence into a relation of
this more simple form. In fact, if we multiply every-
thing by the so-called summing factor:
a
n
a
n−1
. . . a
0
b
n
b
n−1
. . . b
0
provided none of a
n
, a
n−1
, . . . , a
0
, b
n
, b
n−1
, . . . , b
0
is
zero, we obtain:
a
n+1
a
n
. . . a
0
b
n
b
n−1
. . . b
0
f
n+1
=
=
a
n
a
n−1
. . . a
0
b
n−1
b
n−2
. . . b
0
f
n
+
a
n
a
n−1
. . . a
0
b
n
b
n−1
. . . b
0
c
n
.
We can now define:
g
n+1
= a
n+1
a
n
. . . a
0
f
n+1
/(b
n
b
n−1
. . . b
0
),
and the relation becomes:
g
n+1
= g
n
+
a
n
a
n−1
. . . a
0
b
n
b
n−1
. . . b
0
c
n
g
0
= a
0
f
0
.
Finally, by unfolding this recurrence the result is:
f
n+1
=
b
n
b
n−1
. . . b
0
a
n+1
a
n
. . . a
0

a
0
f
0
+
n
¸
k=0
a
k
a
k−1
. . . a
0
b
k
b
k−1
. . . b
0
c
k

.
As a technical remark, we observe that sometimes
a
0
and/or b
0
can be 0; in that case, we can unfold the
50 CHAPTER 4. GENERATING FUNCTIONS
recurrence down to 1, and accordingly change the last
index 0.
In order to show a non-trivial example, let us
discuss the problem of determining the coefficient
of t
n
in the f.p.s. corresponding to the function
f(t) =
√
1 −t ln(1/(1 − t)). If we expand the func-
tion, we find:
f(t) =
√
1 −t ln
1
1 −t
=
= t −
1
24
t
3
−
1
24
t
4
−
71
1920
t
5
−
31
960
t
6
+ .
A method for finding a recurrence relation for the
coefficients f
n
of this f.p.s. is to derive a differential
equation for f(t). By differentiating:
f
′
(t) = −
1
2
√
1 −t
ln
1
1 −t
+
1
√
1 −t
and therefore we have the differential equation:
(1 −t)f
′
(t) = −
1
2
f(t) +
√
1 −t.
By extracting the coefficient of t
n
, we have the rela-
tion:
(n + 1)f
n+1
−nf
n
= −
1
2
f
n
+

1/2
n

(−1)
n
which can be written as:
(n + 1)f
n+1
=
2n −1
2
f
n
−
1
4
n
(2n −1)

2n
n

.
This is a recurrence relation of the first order with the
initial condition f
0
= 0. Let us apply the summing
factor method, for which we have a
n
= n, b
n
= (2n−
1)/2. Since a
0
= 0, we have:
a
n
a
n−1
. . . a
1
b
n
b
n−1
. . . b
1
=
n(n −1) 1 2
n
(2n −1)(2n −3) 1
=
=
n!2
n
2n(2n −2) 2
2n(2n −1)(2n −2) 1
=
=
4
n
n!
2
(2n)!
.
By multiplying the recurrence relation by this sum-
ming factor, we find:
(n + 1)
4
n
n!
2
(2n)!
f
n+1
=
2n −1
2
4
n
n!
2
(2n)!
f
n
−
1
2n −1
.
We are fortunate and c
n
simplifies dramatically;
besides, we know that the two coefficients of f
n+1
and f
n
are equal, notwithstanding their appearance.
Therefore we have:
f
n+1
=
1
(n + 1)4
n

2n
n

n
¸
k=0
−1
2k −1
(a
0
f
0
= 0).
We can somewhat simplify this expression by observ-
ing that:
n
¸
k=0
1
2k −1
=
1
2n −1
+
1
2n −3
+ + 1 −1 =
=
1
2n
+
1
2n −1
+
1
2n −2
+ +
1
2
+1−
1
2n
− −
1
2
−1 =
= H
2n+2
−
1
2n + 1
−
1
2n + 2
−
1
2
H
n+1
+
1
2n + 2
−1 =
= H
2n+2
−
1
2
H
n+1
−
2(n + 1)
2n + 1
.
Furthermore, we have:
1
(n + 1)4
n

2n
n

=
4
(n + 1)4
n+1

2n + 2
n + 1

n + 1
2(2n + 1)
=
2
2n + 1
1
4
n+1

2n + 2
n + 1

.
Therefore:
f
n+1
=

1
2
H
n+1
−H
2n+2
+
2(n + 1)
2n + 1



2
2n + 1
1
4
n+1

2n + 2
n + 1

.
This expression allows us to obtain a formula for f
n
:
f
n
=

1
2
H
n
−H
2n
+
2n
2n −1

2
2n −1
1
4
n

2n
n

=
=

H
n
−2H
2n
+
4n
2n −1

1/2
n

.
The reader can numerically check this expression
against the actual values of f
n
given above. By using
the asymptotic approximation H
n
∼ ln n+γ given in
the Introduction, we find:
H
n
−2H
2n
+
4n
2n −1
∼
∼ ln n +γ −2(ln 2 + lnn +γ) + 2 =
= −ln n −γ −ln 4 + 2.
Besides:
1
2n −1
1
4
n

2n
n

∼
1
2n
√
πn
and we conclude:
f
n
∼ −(ln n +γ + ln 4 −2)
1
2n
√
πn
which shows that [f
n
[ grows as ln n/n
3/2
.
4.11. THE INTERNAL PATH LENGTH OF BINARY TREES 51
4.11 The internal path length
of binary trees
Binary trees are often used as a data structure to re-
trieve information. A set D of keys is given, taken
from an ordered universe U. Therefore D is a permu-
tation of the ordered sequence d
1
< d
2
< < d
n
,
and as the various elements arrive, they are inserted
in a binary tree. As we know, there are n! pos-
sible permutations of the keys in D, but there are
only

2n
n

/(n+1) different binary trees with n nodes.
When we are looking for some key d to find whether
d ∈ D or not, we perform a search in the binary tree,
comparing d against the root and other keys in the
tree, until we find d or we arrive to some leaf and we
cannot go on with our search. In the former case our
search is successful, while in the latter case it is un-
successful. The problem is: how many comparisons
should we perform, on the average, to find out that d
is present in the tree (successful search)? The answer
to this question is very important, because it tells us
how good binary trees are for searching information.
The number of nodes along a path in the tree, start-
ing at the root and arriving to a given node K is
called the internal path length for K. It is just the
number of comparisons necessary to find the key in
K. Therefore, our previous problem can be stated in
the following way: what is the average internal path
length for binary trees with n nodes? Knuth has
found a rather simple way for answering this ques-
tion; however, we wish to show how the method of
generating functions can be used to find the average
internal path length in a standard way. The reason-
ing is as follows: we evaluate the total internal path
length for all the trees generated by the n! possible
permutations of our key set D, and then divide this
number by n!n, the total number of nodes in all the
trees.
A non-empty binary tree can be seen as two sub-
trees connected to the root (see Section 2.9); the left
subtree contains k nodes (k = 0, 1, . . . , n−1) and the
right subtree contains the remaining n−1 −k nodes.
Let P
n
the total internal path length (i.p.l.) of all the
n! possible trees generated by permutations. The left
subtrees have therefore a total i.p.l. equal to P
k
, but
every search in these subtrees has to pass through
the root. This increases the total i.p.l. by the total
number of nodes, i.e., it actually is P
k
+ k!k. We
now observe that every left subtree is associated to
each possible right subtree and therefore it should be
counted (n − 1 − k)! times. Besides, every permuta-
tion generating the left and right subtrees is not to
be counted only once: the keys can be arranged in
all possible ways in the overall permutation, retain-
ing their relative ordering. These possible ways are

n−1
k

, and therefore the total contribution to P
n
of
the left subtrees is:

n −1
k

(n−1 −k)!(P
k
+k!k) = (n−1)!

P
k
k!
+k

.
In a similar way we find the total contribution of the
right subtrees:

n −1
k

k!(P
n−1−k
+ (n −1 −k)!(n −1 −k)) =
= (n −1)!

P
n−1−k
(n −1 −k)!
+ (n −1 −k)

.
It only remains to count the contribution of the roots,
which obviously amounts to n!, a single comparisons
for every tree. We therefore have the following recur-
rence relation, in which the contributions of the left
and right subtrees turn out to be the same:
P
n
= n! + (n −1)!

n−1
¸
k=0

P
k
k!
+k +
P
n−1−k
(n −1 −k)!
+ (n −1 −k)

=
= n! + 2(n −1)!
n−1
¸
k=0

P
k
k!
+k

=
= n! + 2(n −1)!

n−1
¸
k=0
P
k
k!
+
n(n −1)
2

.
We used the formula for the sum of the first n − 1
integers, and now, by dividing by n!, we have:
P
n
n!
= 1 +
2
n
n−1
¸
k=0
P
k
k!
+n −1 = n +
2
n
n−1
¸
k=0
P
k
k!
.
Let us now set Q
n
= P
n
/n!, so that Q
n
is the average
total i.p.l. relative to a single tree. If we’ll succeed
in finding Q
n
, the average i.p.l. we are looking for
will simply be Q
n
/n. We can also reformulate the
recurrence for n +1, in order to be able to apply the
generating function operator:
(n + 1)Q
n+1
= (n + 1)
2
+ 2
n
¸
k=0
Q
k
Q
′
(t) =
1 +t
(1 −t)
3
+ 2
Q(t)
1 −t
Q
′
(t) −
2
1 −t
Q(t) =
1 +t
(1 −t)
3
.
This differential equation can be easily solved:
Q(t) =
1
(1 −t)
2

(1 −t)
2
(1 +t)
(1 −t)
3
dt +C

=
=
1
(1 −t)
2

2 ln
1
1 −t
−t +C

.
52 CHAPTER 4. GENERATING FUNCTIONS
Since the i.p.l. of the empty tree is 0, we should have
Q
0
= Q(0) = 0 and therefore, by setting t = 0, we
find C = 0. The final result is:
Q(t) =
2
(1 −t)
2
ln
1
1 −t
−
t
(1 −t)
2
.
We can now use the formula for ((nH
n
) (see the Sec-
tion 4.4 on Common Generating Functions) to ex-
tract the coefficient of t
n
:
Q
n
= [t
n
]
1
(1 −t)
2

2

ln
1
1 −t
+ 1

−(2 +t)

=
= 2[t
n+1
]
t
(1 −t)
2

ln
1
1 −t
+ 1

−[t
n
]
2 +t
(1 −t)
2
=
= 2(n+1)H
n+1
−2

−2
n

(−1)
n
−

−2
n −1

(−1)
n−1
=
= 2(n +1)H
n
+2 −2(n +1) −n = 2(n +1)H
n
−3n.
Thus we conclude with the average i.p.l.:
P
n
n!n
=
Q
n
n
= 2

1 +
1
n

H
n
−3.
This formula is asymptotic to 2 ln n+γ−3, and shows
that the average number of comparisons necessary to
retrieve any key in a binary tree is in the order of
O(ln n).
4.12 Height balanced binary
trees
We have been able to show that binary trees are a
“good” retrieving structure, in the sense that if the
elements, or keys, of a set ¦a
1
, a
2
, . . . , a
n
¦ are stored
in random order in a binary (search) tree, then the
expected average time for retrieving any key in the
tree is in the order of lnn. However, this behavior of
binary trees is not always assured; for example, if the
keys are stored in the tree in their proper order, the
resulting structure degenerates into a linear list and
the average retrieving time becomes O(n).
To avoid this drawback, at the beginning of the
1960’s, two Russian researchers, Adelson-Velski and
Landis, found an algorithm to store keys in a “height
balanced” binary tree, a tree for which the height of
the left subtree of every node K differs by at most
1 from the height of the right subtree of the same
node K. To understand this concept, let us define the
height of a tree as the highest level at which a node
in the tree is placed. The height is also the maximal
number of comparisons necessary to find any key in
the tree. Therefore, if we find a limitation for the
height of a class of trees, this is also a limitation for
the internal path length of the trees in the same class.
Formally, a height balanced binary tree is a tree such
that for every node K in it, if h
′
K
and h
′′
k
are the
heights of the two subtrees originating from K, then
[h
′
K
−h
′′
K
[ ≤ 1.
The algorithm of Adelson-Velski and Landis is very
important because, as we are now going to show,
height balanced binary trees assure that the retriev-
ing time for every key in the tree is O(ln n). Because
of that, height balanced binary trees are also known
as AVL trees, and the algorithm for building AVL-
trees from a set of n keys can be found in any book
on algorithms and data structures. Here we only wish
to perform a worst case analysis to prove that the re-
trieval time in any AVL tree cannot be larger than
O(ln n).
In order to perform our analysis, let us consider
to worst possible AVL tree. Since, by definition, the
height of the left subtree of any node cannot exceed
the height of the corresponding right subtree plus 1,
let us consider trees in which the height of the left
subtree of every node exceeds exactly by 1 the height
of the right subtree of the same node. In Figure 4.1
we have drawn the first cases. These trees are built in
a very simple way: every tree T
n
, of height n, is built
by using the preceding tree T
n−1
as the left subtree
and the tree T
n−2
as the right subtree of the root.
Therefore, the number of nodes in T
n
is just the sum
of the nodes in T
n−1
and in T
n−2
, plus 1 (the root),
and the condition on the heights of the subtrees of
every node is satisfied. Because of this construction,
T
n
can be considered as the “worst” tree of height n,
in the sense that every other AVL-tree of height n will
have at least as many nodes as T
n
. Since the height
n is an upper bound for the number of comparisons
necessary to retrieve any key in the tree, the average
retrieving time for every such tree will be ≤ n.
If we denote by [T
n
[ the number of nodes in the
tree T
n
, we have the simple recurrence relation:
[T
n
[ = [T
n−1
[ +[T
n−2
[ + 1.
This resembles the Fibonacci recurrence relation,
and, in fact, we can easily show that [T
n
[ = F
n+1
−1,
as is intuitively apparent from the beginning of the
sequence ¦0, 1, 2, 4, 7, 12, . . .¦. The proof is done by
mathematical induction. For n = 0 we have [T
0
[ =
F
1
− 1 = 1 − 1 = 0, and this is true; similarly we
proceed for n + 1. Therefore, let us suppose that for
every k < n we have [T
k+1
[ = F
k
− 1; this holds for
k = n−1 and k = n−2, and because of the recurrence
relation for [T
n
[ we have:
[T
n
[ = [T
n−1
[ +[T
n−2
[ + 1 =
= F
n
−1 +F
n−1
−1 + 1 = F
n+1
−1
since F
n
+F
n−1
= F
n+1
by the Fibonacci recurrence.
4.13. SOME SPECIAL RECURRENCES 53
T
0
T
1
T
2
T
3
T
4
T
5
,
,
,


,
,
,
,



d
d
,
,
,
,
,
,
,






e
e
d
d
¡
¡
,
,
,
,
,
,
,
,
,
,
,
,







e
e
d
d
¡
¡
d
d
d
d
¡
¡
¡
¡
Figure 4.1: Worst AVL-trees
We have shown that for large values of n we have
F
n
≈ φ
n
/
√
5; therefore we have [T
n
[ ≈ φ
n+1
/
√
5 −1
or φ
n+1
≈
√
5([T
n
[+1). By passing to the logarithms,
we have: n ≈ log
φ
(
√
5([T
n
[ + 1)) − 1, and since all
logarithms are proportional, n = O(ln [T
n
[). As we
observed, every AVL-tree of height n has a number
of nodes not less than [T
n
[, and this assures that the
retrieving time for every AVL-tree with at most [T
n
[
nodes is bounded from above by log
φ
(
√
5([T
n
[+1))−1
4.13 Some special recurrences
Not all recurrence relations are linear and we had
occasions to deal with a different sort of relation when
we studied the Catalan numbers. They satisfy the
recurrence C
n
=
¸
n−1
k=0
C
k
C
n−k−1
, which however,
in this particular form, is only valid for n > 0. In
order to apply the method of generating functions,
we write it for n + 1:
C
n+1
=
n
¸
k=0
C
k
C
n−k
.
The right hand member is a convolution, and there-
fore, by the initial condition C
0
= 1, we obtain:
C(t) −1
t
= C(t)
2
or tC(t)
2
−C(t) + 1 = 0.
This is a second degree equation, which can be di-
rectly solved; for t = 0 we should have C(0) = C
0
=
1, and therefore the solution with the + sign before
the square root is to be ignored; we thus obtain:
C(t) =
1 −
√
1 −4t
2t
which was found in the section “The method of shift-
ing” in a completely different way.
The Bernoulli numbers were introduced by means
of the implicit relation:
n
¸
k=0

n + 1
k

B
k
= δ
n,0
.
We are now in a position to find out their expo-
nential generating function, i.e., the function B(t) =
((B
n
/n!), and prove some of their properties. The
defining relation can be written as:
n
¸
k=0

n + 1
n −k

B
n−k
=
=
n
¸
k=0
(n + 1)n (k + 2)
B
n−k
(n −k)!
=
=
n
¸
k=0
(n + 1)!
(k + 1)!
B
n−k
(n −k)!
= δ
n,0
.
If we divide everything by (n + 1)!, we obtain:
n
¸
k=0
1
(k + 1)!
B
n−k
(n −k)!
= δ
n,0
and since this relation holds for every n ∈ N, we
can pass to the generating functions. The left hand
member is a convolution, whose first factor is the shift
of the exponential function and therefore we obtain:
e
t
−1
t
B(t) = 1 or B(t) =
t
e
t
−1
.
The classical way to see that B
2n+1
= 0, ∀n > 0,
is to show that the function obtained from B(t) by
deleting the term of first degree is an even function,
and therefore should have all its coefficients of odd
order equal to zero. In fact we have:
B(t) +
t
2
=
t
e
t
−1
+
t
2
=
t
2
e
t
+ 1
e
t
−1
.
In order to see that this function is even, we substi-
tute t → −t and show that the function remains the
same:
−
t
2
e
−t
+ 1
e
−t
−1
= −
t
2
1 +e
t
1 −e
t
=
t
2
e
t
+ 1
e
t
−1
.
54 CHAPTER 4. GENERATING FUNCTIONS
Chapter 5
Riordan Arrays
5.1 Definitions and basic con-
cepts
A Riordan array is a couple of formal power series
D = (d(t), h(t)); if both d(t), h(t) ∈ T
0
, then the
Riordan array is called proper. The Riordan array
can be identified with the infinite, lower triangular
array (or triangle) (d
n,k
)
n,k∈N
defined by:
d
n,k
= [t
n
]d(t)(th(t))
k
(5.1.1)
In fact, we are mainly interested in the sequence of
functions iteratively defined by:

d
0
(t) = d(t)
d
k
(t) = d
k−1
(t)th(t) = d(t)(th(t))
k
These functions are the column generating functions
of the triangle.
Another way of characterizing a Riordan array
D = (d(t), h(t)) is to consider the bivariate gener-
ating function:
d(t, z) =
∞
¸
k=0
d(t)(th(t))
k
z
k
=
d(t)
1 −tzh(t)
(5.1.2)
A common example of a Riordan array is the Pascal
triangle, for which we have d(t) = h(t) = 1/(1 − t).
In fact we have:
d
n,k
= [t
n
]
1
1 −t

t
1 −t

k
= [t
n−k
]
1
(1 −t)
k+1
=
=

−k −1
n −k

(−1)
n−k
=

n
n −k

=

n
k

and this shows that the generic element is actually a
binomial coefficient. By formula (5.1.2), we find the
well-known bivariate generating function d(t, z) =
(1 −t −tz)
−1
.
From our point of view, one of the most impor-
tant properties of Riordan arrays is the fact that the
sums involving the rows of a Riordan array can be
performed by operating a suitable transformation on
a generating function and then by extracting a coeffi-
cient from the resulting function. More precisely, we
prove the following theorem:
Theorem 5.1.1 Let D = (d(t), h(t)) be a Riordan
array and let f(t) be the generating function of the
sequence (f
k
)
k∈N
; then we have:
n
¸
k=0
d
n,k
f
k
= [t
n
]d(t)f(th(t)) (5.1.3)
Proof: The proof consists in a straight-forward com-
putation:
n
¸
k=0
d
n,k
f
k
=
∞
¸
k=0
d
n,k
f
k
=
=
∞
¸
k=0
[t
n
]d(t)(th(t))
k
f
k
=
= [t
n
]d(t)
∞
¸
k=0
f
k
(th(t))
k
=
= [t
n
]d(t)f(th(t)).
In the case of Pascal triangle we obtain the Euler
transformation:
n
¸
k=0

n
k

f
k
= [t
n
]
1
1 −t
f

t
1 −t

which we proved by simple considerations on bino-
mial coefficients and generating functions. By con-
sidering the generating functions ((1) = 1/(1 − t),
(

(−1)
k

= 1/(1+t) and ((k) = t/(1−t)
2
, from the
previous theorem we immediately find:
row sums (r. s.)
¸
k
d
n,k
= [t
n
]
d(t)
1 −th(t)
alternating r. s.
¸
k
(−1)
k
d
n,k
= [t
n
]
d(t)
1 +th(t)
weighted r. s.
¸
k
kd
n,k
= [t
n
]
td(t)h(t)
(1 −th(t))
2
.
Moreover, by observing that
´
D = (d(t), th(t)) is a
Riordan array, whose rows are the diagonals of D,
55
56 CHAPTER 5. RIORDAN ARRAYS
we have:
diagonal sums
¸
k
d
n−k,k
= [t
n
]
d(t)
1 −t
2
h(t)
.
Obviously, this observation can be generalized to find
the generating function of any sum
¸
k
d
n−sk,k
for
every s ≥ 1. We obtain well-known results for the
Pascal triangle; for example, diagonal sums give:
¸
k

n −k
k

= [t
n
]
1
1 −t
1
1 −t
2
(1 −t)
−1
=
= [t
n
]
1
1 −t −t
2
= F
n+1
connecting binomial coefficients and Fibonacci num-
bers.
Another general result can be obtained by means of
two sequences (f
k
)
k∈N
and (g
k
)
k∈N
and their gener-
ating functions f(t), g(t). For p = 1, 2, . . ., the generic
element of the Riordan array (f(t), t
p−1
) is:
d
n,k
= [t
n
]f(t)(t
p
)
k
= [t
n−pk
]f(t) = f
n−pk
.
Therefore, by formula (5.1.3), we have:
⌊n/p⌋
¸
k=0
f
n−pk
g
k
= [t
n
]f(t)

g(y)


y = t
p

=
= [t
n
]f(t)g(t
p
).
This can be called the rule of generalized convolution
since it reduces to the usual convolution rule for p =
1. Suppose, for example, that we wish to sum one
out of every three powers of 2, starting with 2
n
and
going down to the lowest integer exponent ≥ 0; we
have:
S
n
=
⌊n/3⌋
¸
k=0
2
n−3k
= [t
n
]
1
1 −2t
1
1 −t
3
.
As we will learn studying asymptotics, an approxi-
mate value for this sum can be obtained by extracting
the coefficient of the first factor and then by multiply-
ing it by the second factor, in which t is substituted
by 1/2. This gives S
n
≈ 2
n+3
/7, and in fact we have
the exact value S
n
= ⌊2
n+3
/7⌋.
In a sense, the theorem on the sums involving the
Riordan arrays is a characterization for them; in fact,
we can prove a sort of inverse property:
Theorem 5.1.2 Let (d
n,k
)
n,k∈N
be an infinite tri-
angle such that for every sequence (f
k
)
k∈N
we have
¸
k
d
n,k
f
k
= [t
n
]d(t)f(th(t)), where f(t) is the gen-
erating function of the sequence and d(t), h(t) are two
f.p.s. not depending on f(t). Then the triangle de-
fined by the Riordan array (d(t), h(t)) coincides with
(d
n,k
)
n,k∈N
.
Proof: For every k ∈ N take the sequence which
is 0 everywhere except in the kth element f
k
= 1.
The corresponding generating function is f(t) = t
k
and we have
¸
∞
i=0
d
n,i
f
i
= d
n,k
. Hence, according to
the theorem’s hypotheses, we find (
t
(d
n,k
)
n,k∈N
=
d
k
(t) = d(t)(th(t))
k
, and this corresponds to the ini-
tial definition of column generating functions for a
Riordan array, for every k = 1, 2, . . ..
5.2 The algebraic structure of
Riordan arrays
The most important algebraic property of Riordan
arrays is the fact that the usual row-by-column prod-
uct of two Riordan arrays is a Riordan array. This is
proved by considering two Riordan arrays (d(t), h(t))
and (a(t), b(t)) and performing the product, whose
generic element is
¸
j
d
n,j
f
j,k
, if d
n,j
is the generic
element in (d(t), h(t)) and f
j,k
is the generic element
in (a(t), b(t)). In fact we have:
∞
¸
j=0
d
n,j
f
j,k
=
=
∞
¸
j=0
[t
n
]d(t)(th(t))
j
[y
j
]a(y)(yb(y))
k
=
= [t
n
]d(t)
∞
¸
j=0
(th(t))
j
[y
j
]a(y)(yb(y))
k
=
= [t
n
]d(t)a(th(t))(th(t)b(th(t)))
k
.
By definition, the last expression denotes the generic
element of the Riordan array (f(t), g(t)) where f(t) =
d(t)a(th(t)) and g(t) = h(t)b(th(t)). Therefore we
have:
(d(t), h(t)) (a(t), b(t)) = (d(t)a(th(t)), h(t)b(th(t))).
(5.2.1)
This expression is particularly important and is the
basis for many developments of the Riordan array
theory.
The product is obviously associative, and we ob-
serve that the Riordan array (1, 1) acts as the neutral
element or identity. In fact, the array (1, 1) is every-
where 0 except for the elements on the main diagonal,
which are 1. Observe that this array is proper.
Let us now suppose that (d(t), h(t)) is a proper
Riordan array. By formula (5.2.1), we immediately
see that the product of two proper Riordan arrays is
proper; therefore, we can look for a proper Riordan
array (a(t), b(t)) such that (d(t), h(t)) (a(t), b(t)) =
(1, 1). If this is the case, we should have:
d(t)a(th(t)) = 1 and h(t)b(th(t)) = 1.
5.3. THE A-SEQUENCE FOR PROPER RIORDAN ARRAYS 57
By setting y = th(t) we have:
a(y) =

d(t)
−1


t = yh(t)
−1

b(y) =

h(t)
−1


t = yh(t)
−1

.
Here we are in the hypotheses of the Lagrange Inver-
sion Formula, and therefore there is a unique function
t = t(y) such that t(0) = 0 and t = yh(t)
−1
. Besides,
being d(t), h(t) ∈ T
0
, the two f.p.s. a(y) and b(y) are
uniquely defined. We have therefore proved:
Theorem 5.2.1 The set / of proper Riordan arrays
is a group with the operation of row-by-column prod-
uct defined functionally by relation (5.2.1).
It is a simple matter to show that some important
classes of Riordan arrays are subgroups of /:
• the set of the Riordan arrays (f(t), 1) is an in-
variant subgroup of /; it is called the Appell
subgroup;
• the set of the Riordan arrays (1, g(t)) is a sub-
group of / and is called the subgroup of associ-
ated operators or the Lagrange subgroup;
• the set of Riordan arrays (f(t), f(t)) is a sub-
group of / and is called the Bell subgroup. Its
elements are also known as renewal arrays.
The first two subgroups have already been consid-
ered in the Chapter on “Formal Power Series” and
show the connection between f.p.s. and Riordan ar-
rays. The notations used in that Chapter are thus
explained as particular cases of the most general case
of (proper) Riordan arrays.
Let us now return to the formulas for a Riordan
array inverse. If h(t) is any fixed invertible f.p.s., let
us define:
d
h
(t) =

d(y)
−1


y = th(y)
−1

so that we can write (d(t), h(t))
−1
= (d
h
(t), h
h
(t)).
By the product formula (5.2.1) we immediately find
the identities:
d(th
h
(t)) = d
h
(t)
−1
d
h
(th(t)) = d(t)
−1
h(th
h
(t)) = h
h
(t)
−1
h
h
(th(t)) = h(t)
−1
which can be reduced to the single and basic rule:
f(th
h
(t)) = f
h
(t)
−1
∀f(t) ∈ T
0
.
Observe that obviously f
h
(t) = f(t).
We wish now to find an explicit expression for the
generic element d
n,k
in the inverse Riordan array
(d(t), h(t))
−1
in terms of d(t) and h(t). This will be
done by using the LIF. As we observed in the first sec-
tion, the bivariate generating function for (d(t), h(t))
is d(t)/(1 −tzh(t)) and therefore we have:
d
n,k
= [t
n
z
k
]
d
h
(t)
1 −tzh
h
(t)
=
= [z
k
][t
n
]
¸
d
h
(t)
1 −zy


y = th
h
(t)

.
By the formulas above, we have:
y = th
h
(t) = th(th
h
(t))
−1
= th(y)
−1
which is the same as t = yh(y). Therefore we find:
d
h
(t) = d
h
(yh(y)) = d(t)
−1
, and consequently:
d
n,k
= [z
k
][t
n
]
¸
d(y)
−1
1 −zy


y = th(y)
−1

=
= [z
k
]
1
n
[y
n−1
]

d
dy
d(y)
−1
1 −zy

1
h(y)
n
=
= [z
k
]
1
n
[y
n−1
]

z
d(y)(1 −zy)
2
−
−
d
′
(y)
d(y)
2
(1 −zy)

1
h(y)
n
=
= [z
k
]
1
n
[y
n−1
]

∞
¸
r=0
z
r+1
y
r
(r + 1)−
−
d
′
(y)
d(y)
∞
¸
r=0
z
r
y
r

1
d(y)h(y)
n
=
=
1
n
[y
n−1
]

ky
k−1
−y
k
d
′
(y)
d(y)

1
d(y)h(y)
n
=
=
1
n
[y
n−k
]

k −
yd
′
(y)
d(y)

1
d(y)h(y)
n
.
This is the formula we were looking for.
5.3 The A-sequence for proper
Riordan arrays
Proper Riordan arrays play a very important role
in our approach. Let us consider a Riordan array
D = (d(t), h(t)), which is not proper, but d(t) ∈
T
0
. Since h(0) = 0, an s > 0 exists such that
h(t) = h
s
t
s
+h
s+1
t
s+1
+ and h
s
= 0. If we define
´
h(t) = h
s
+h
s+1
t+ , then
´
h(t) ∈ T
0
. Consequently,
the Riordan array
´
D = (d(t),
´
h(t)) is proper and the
rows of D can be seen as the s-diagonals (
´
d
n−sk
)
k∈N
of
´
D. Fortunately, for proper Riordan arrays, Rogers
has found an important characterization: every ele-
ment d
n+1,k+1
, n, k ∈ N, can be expressed as a linear
combination of the elements in the preceding row,
i.e.:
d
n+1,k+1
= a
0
d
n,k
+a
1
d
n,k+1
+a
2
d
n,k+2
+ =
58 CHAPTER 5. RIORDAN ARRAYS
=
∞
¸
j=0
a
j
d
n,k+j
. (5.3.1)
The sum is actually finite and the sequence A =
(a
k
)
k∈N
is fixed. More precisely, we can prove the
following theorem:
Theorem 5.3.1 An infinite lower triangular array
D = (d
n,k
)
n,k∈N
is a Riordan array if and only if a
sequence A = ¦a
0
= 0, a
1
, a
2
, . . .¦ exists such that for
every n, k ∈ N relation (5.3.1) holds
Proof: Let us suppose that D is the Riordan
array (d(t), h(t)) and let us consider the Riordan
array (d(t)h(t), h(t)); we define the Riordan array
(A(t), B(t)) by the relation:
(A(t), B(t)) = (d(t), h(t))
−1
(d(t)h(t), h(t))
or:
(d(t), h(t)) (A(t), B(t)) = (d(t)h(t), h(t)).
By performing the product we find:
d(t)A(th(t)) = d(t)h(t) and h(t)B(th(t)) = h(t).
The latter identity gives B(th(t)) = 1 and this implies
B(t) = 1. Therefore we have (d(t), h(t)) (A(t), 1) =
(d(t)h(t), h(t)). The element f
n,k
of the left hand
member is
¸
∞
j=0
d
n,j
a
k−j
=
¸
∞
j=0
d
n,k+j
a
j
, if as
usual we interpret a
k−j
as 0 when k < j. The same
element in the right hand member is:
[t
n
]d(t)h(t)(th(t))
k
=
= [t
n+1
]d(t)(th(t))
k+1
= d
n+1,k+1
.
By equating these two quantities, we have the iden-
tity (5.3.1). For the converse, let us observe that
(5.3.1) uniquely defines the array D when the ele-
ments ¦d
0,0
, d
1,0
, d
2,0
, . . .¦ of column 0 are given. Let
d(t) be the generating function of this column, A(t)
the generating function of the sequence A and de-
fine h(t) as the solution of the functional equation
h(t) = A(th(t)), which is uniquely determined be-
cause of our hypothesis a
0
= 0. We can therefore
consider the proper Riordan array
´
D = (d(t), h(t));
by the first part of the theorem,
´
D satisfies relation
(5.3.1), for every n, k ∈ N and therefore, by our previ-
ous observation, it must coincide with D. This com-
pletes the proof.
The sequence A = (a
k
)
k∈N
is called the A-sequence
of the Riordan array D = (d(t), h(t)) and it only
depends on h(t). In fact, as we have shown during
the proof of the theorem, we have:
h(t) = A(th(t)) (5.3.2)
and this uniquely determines A when h(t) is given
and, vice versa, h(t) is uniquely determined when A
is given.
The A-sequence for the Pascal triangle is the so-
lution A(y) of the functional equation 1/(1 − t) =
A(t/(1 − t)). The simple substitution y = t/(1 − t)
gives A(y) = 1 +y, corresponding to the well-known
basic recurrence of the Pascal triangle:

n+1
k+1

=

n
k

+

n
k+1

. At this point, we realize that we could
have started with this recurrence relation and directly
found A(y) = 1+y. Now, h(t) is defined by (5.3.2) as
the solution of h(t) = 1+th(t), and this immediately
gives h(t) = 1/(1−t). Furthermore, since column 0 is
¦1, 1, 1, . . .¦, we have proved that the Pascal triangle
corresponds to the Riordan array (1/(1−t), 1/(1−t))
as initially stated.
The pair of functions d(t) and A(t) completely
characterize a proper Riordan array. Another type
of characterization is obtained through the following
observation:
Theorem 5.3.2 Let (d
n,k
)
n,k∈N
be any infinite,
lower triangular array with d
n,n
= 0, ∀n ∈ N (in
particular, let it be a proper Riordan array); then a
unique sequence Z = (z
k
)
k∈N
exists such that every
element in column 0 can be expressed as a linear com-
bination of all the elements in the preceding row, i.e.:
d
n+1,0
= z
0
d
n,0
+z
1
d
n,1
+z
2
d
n,2
+ =
∞
¸
j=0
z
j
d
n,j
.
(5.3.3)
Proof: Let z
0
= d
1,0
/d
0,0
. Now we can uniquely
determine the value of z
1
by expressing d
2,0
in terms
of the elements in row 1, i.e.:
d
2,0
= z
0
d
1,0
+z
1
d
1,1
or z
1
=
d
0,0
d
2,0
−d
2
1,0
d
0,0
d
1,1
.
In the same way, we determine z
2
by expressing d
3,0
in terms of the elements in row 2, and by substituting
the values just obtained for z
0
and z
1
. By proceeding
in the same way, we determine the sequence Z in a
unique way.
The sequence Z is called the Z-sequence for the
(Riordan) array; it characterizes column 0, except
for the element d
0,0
. Therefore, we can say that
the triple (d
0,0
, A(t), Z(t)) completely characterizes
a proper Riordan array. To see how the Z-sequence
is obtained by starting with the usual definition of a
Riordan array, let us prove the following:
Theorem 5.3.3 Let (d(t), h(t)) be a proper Riordan
array and let Z(t) be the generating function of the
array’s Z-sequence. We have:
d(t) =
d
0,0
1 −tZ(th(t))
5.4. SIMPLE BINOMIAL COEFFICIENTS 59
Proof: By the preceding theorem, the Z-sequence
exists and is unique. Therefore, equation (5.3.3) is
valid for every n ∈ N, and we can go on to the gener-
ating functions. Since d(t)(th(t))
k
is the generating
function for column k, we have:
d(t) −d
0,0
t
=
= z
0
d(t) +z
1
d(t)th(t) +z
2
d(t)(th(t))
2
+ =
= d(t)(z
0
+z
1
th(t) +z
2
(th(t))
2
+ ) =
= d(t)Z(th(t)).
By solving this equation in d(t), we immediately find
the relation desired.
The relation can be inverted and this gives us the
formula for the Z-sequence:
Z(y) =
¸
d(t) −d
0,0
td(t)


t = yh(t)
−1

.
We conclude this section by giving a theorem,
which characterizes renewal arrays by means of the
A- and Z-sequences:
Theorem 5.3.4 Let d(0) = h(0) = 0. Then d(t) =
h(t) if and only if: A(y) = d(0) +yZ(y).
Proof: Let us assume that A(y) = d(0) + yZ(y) or
Z(y) = (A(y) − d(0))/y. By the previous theorem,
we have:
d(t) =
d(0)
1 −tZ(th(t))
=
=
d(0)
1 −(tA(th(t)) −d(0)t)/th(t)
=
=
d(0)th(t)
d(0)t
= h(t),
because A(th(t)) = h(t) by formula (5.3.2). Vice
versa, by the formula for Z(y), we obtain from the
hypothesis d(t) = h(t):
d(0) +yZ(y) =
=
¸
d(0) +y

1
t
−
d(0)
th(t)



t = yh(t)
−1

=
=
¸
d(0) +
th(t)
t
−
d(0)th(t)
th(t)


t = yh(t)
−1

=
=

h(t)


t = yh(t)
−1

= A(y).
5.4 Simple binomial coeffi-
cients
Let us consider simple binomial coefficients, i.e., bi-
nomial coefficients of the form

n+ak
m+bk

, where a, b are
two parameters and k is a non-negative integer vari-
able. Depending if we consider n a variable and m a
parameter, or vice versa, we have two different infi-
nite arrays (d
n,k
) or (
´
d
m,k
), whose elements depend
on the parameters a, b, m or a, b, n, respectively. In
either case, if some conditions on a, b hold, we have
Riordan arrays and therefore we can apply formula
(5.1.3) to find the value of many sums.
Theorem 5.4.1 Let d
n,k
and
´
d
m,k
be as above. If
b > a and b − a is an integer, then D = (d
n,k
) is
a Riordan array. If b < 0 is an integer, then
´
D =
(
´
d
m,k
) is a Riordan array. We have:
D =

t
m
(1 −t)
m+1
,
t
b−a−1
(1 −t)
b

´
D =

(1 +t)
n
,
t
−b−1
(1 +t)
−a

.
Proof: By using well-known properties of binomial
coefficients, we find:
d
n,k
=

n +ak
m+bk

=

n +ak
n −m+ak −bk

=
=

−n −ak +n −m+ak −bk −1
n −m+ak −bk


(−1)
n−m+ak−bk
=
=

−m−bk −1
(n −m) + (a −b)k

(−1)
n−m+ak−bk
=
= [t
n−m+ak−bk
]
1
(1 −t)
m+1+bk
=
= [t
n
]
t
m
(1 −t)
m+1

t
b−a
(1 −t)
b

k
;
and:
´
d
m,k
= [t
m+bk
](1 +t)
n+ak
=
= [t
m
](1 +t)
n
(t
−b
(1 +t)
a
)
k
.
The theorem now directly follows from (5.1.1)
For m = a = 0 and b = 1 we again find the Riordan
array of the Pascal triangle. The sum (5.1.3) takes
on two specific forms which are worth being stated
explicitly:
¸
k

n +ak
m+bk

f
k
=
= [t
n
]
t
m
(1 −t)
m+1
f

t
b−a
(1 −t)
b

b > a (5.4.1)
¸
k

n +ak
m+bk

f
k
=
= [t
m
](1 +t)
n
f(t
−b
(1 +t)
a
) b < 0. (5.4.2)
60 CHAPTER 5. RIORDAN ARRAYS
If m and n are independent of each other, these
relations can also be stated as generating function
identities. The binomial coefficient

n+ak
m+bk

is so gen-
eral that a large number of combinatorial sums can
be solved by means of the two formulas (5.4.1) and
(5.4.2).
Let us begin our set of examples with a simple sum;
by the theorem above, the binomial coefficients

n−k
m

corresponds to the Riordan array (t
m
/(1 +t)
m+1
, 1);
therefore, by the formula concerning the row sums,
we have:
¸
k

n −k
m

= [t
n
]
t
m
(1 −t)
m+1
1
1 −t
=
= [t
n−m
]
1
(1 −t)
m+2
=

n + 1
m+ 1

.
Another simple example is the sum:
¸
k

n
2k + 1

5
k
=
= [t
n
]
t
(1 −t)
2
¸
1
1 −5y


y =
t
2
(1 −t)
2

=
=
1
2
[t
n
]
2t
1 −2t −4t
2
= 2
n−1
F
n
.
The following sum is a more interesting case. From
the generating function of the Catalan numbers we
immediately find:
¸
k

n +k
m+ 2k

2k
k

(−1)
k
k + 1
=
= [t
n
]
t
m
(1 −t)
m+1
¸√
1 + 4y −1
2y


y =
t
(1 −t)
2

=
= [t
n−m
]
1
(1 −t)
m+1

1 +
4t
(1 −t)
2
−1



(1 −t)
2
2t
=
= [t
n−m
]
1
(1 −t)
m
=

n −1
m−1

.
In the following sum we use the bisection formulas.
Because the generating function for

z+1
k

2
k
is (1 +
2t)
z+1
, we have:
(

z + 1
2k + 1

2
2k+1

=
=
1
2
√
t

(1 + 2
√
t)
z+1
−(1 −2
√
t)
z+1

.
By applying formula (5.4.2):
¸
k

z + 1
2k + 1

z −2k
n −k

2
2k+1
=
= [t
n
](1 +t)
z
¸
(1 + 2
√
y)
z+1
2
√
y
−
−
(1 −2
√
y)
z+1
2
√
y


y =
t
(1 +t)
2

=
= [t
n
](1 +t)
z+1

(1 +t + 2
√
t)
z+1
2
√
t(1 +t)
z+1
−
−
(1 +t −2
√
t)
z+1
2
√
t(1 +t)
z+1

=
= [t
2n+1
](1 +t)
2z+2
=

2z + 2
2n + 1

;
in the last but one passage, we used backwards the
bisection rule, since (1 +t ±2
√
t)
z+1
= (1 ±
√
t)
2z+2
.
We solve the following sum by using (5.4.2):
¸
k

2n −2k
m−k

n
k

(−2)
k
=
= [t
m
](1 +t)
2n
¸
(1 −2y)
n


y =
t
(1 −t)
2

=
= [t
m
](1 +t
2
)
n
=

n
m/2

where the binomial coefficient is to be taken as zero
when m is odd.
5.5 Other Riordan arrays from
binomial coefficients
Other Riordan arrays can be found by using the the-
orem in the previous section and the rule (α = 0, if
− is considered):
α ±β
α

α
β

=

α
β

±

α −1
β −1

.
For example we find:
2n
n +k

n +k
2k

=
2n
n +k

n +k
n −k

=
=

n +k
n −k

+

n +k −1
n −k −1

=
=

n +k
2k

+

n −1 +k
2k

.
Hence, by formula (5.4.1), we have:
¸
k
2n
n +k

n +k
n −k

f
k
=
=
¸
k

n +k
2k

f
k
+
¸
k

n −1 +k
2k

f
k
=
= [t
n
]
1
1 −t
f

t
(1 −t)
2

+
+ [t
n−1
]
1
1 −t
f

t
(1 −t)
2

=
= [t
n
]
1 +t
1 −t
f

t
(1 −t)
2

.
5.6. BINOMIAL COEFFICIENTS AND THE LIF 61
This proves that the infinite triangle of the elements
2n
n+k

n+k
2k

is a proper Riordan array and many identi-
ties can be proved by means of the previous formula.
For example:
¸
k
2n
n +k

n +k
n −k

2k
k

(−1)
k
=
= [t
n
]
1 +t
1 −t
¸
1
√
1 + 4y


y =
t
(1 −t)
2

=
= [t
n
]1 = δ
n,0
,
¸
k
2n
n +k

n +k
n −k

2k
k

(−1)
k
k + 1
=
= [t
n
]
1 +t
1 −t
¸√
1 + 4y −1
2y


y =
t
(1 −t)
2

=
= [t
n
](1 +t) = δ
n,0
+δ
n,1
.
The following is a quite different case. Let f(t) =
((f
k
) and:
G(t) = (

f
k
k

=

t
0
f(τ) −f
0
τ
dτ.
Obviously we have:

n −k
k

=
n −k
k

n −k −1
k −1

except for k = 0, when the left-hand side is 1 and the
right-hand side is not defined. By formula (5.4.1):
¸
k
n
n −k

n −k
k

f
k
=
= f
0
+n
∞
¸
k=1

n −k −1
k −1

f
k
k
=
= f
0
+n[t
n
]G

t
2
1 −t

. (5.5.1)
This gives an immediate proof of the following for-
mula known as Hardy’s identity:
¸
k
n
n −k

n −k
k

(−1)
k
=
= [t
n
] ln
1 −t
2
1 +t
3
=
= [t
n
]

ln
1
1 +t
3
−ln
1
1 −t
2

=
=

(−1)
n
2/n if 3 divides n
(−1)
n−1
/n otherwise.
We also immediately obtain:
¸
k
1
n −k

n −k
k

=
φ
n
+
´
φ
n
n
where φ is the golden ratio and
´
φ = −φ
−1
. The
reader can generalize formula (5.5.1) by using the
change of variable t → pt and prove other formulas.
The following one is known as Riordan’s old identity:
¸
k
n
n −k

n −k
k

(a +b)
n−2k
(−ab)
k
= a
n
+b
n
while this is a generalization of Hardy’s identity:
¸
k
n
n −k

n −k
k

x
n−2k
(−1)
k
=
=
(x +
√
x
2
−4)
n
+ (x −
√
x
2
−4)
n
2
n
.
5.6 Binomial coefficients and
the LIF
In a few cases only, the formulas of the previous sec-
tions give the desired result when the m and n in
the numerator and denominator of a binomial coeffi-
cient are related between them. In fact, in that case,
we have to extract the coefficient of t
n
from a func-
tion depending on the same variable n (or m). This
requires to apply the Lagrange Inversion Formula, ac-
cording to the diagonalization rule. Let us suppose
we have the binomial coefficient

2n−k
n−k

and we wish
to know whether it corresponds to a Riordan array
or not. We have:

2n −k
n −k

= [t
n−k
](1 +t)
2n−k
=
= [t
n
](1 +t)
2n

t
1 +t

k
.
The function (1 + t)
2n
cannot be assumed as the
d(t) function of a Riordan array because it varies
as n varies. Therefore, let us suppose that k is
fixed; we can apply the diagonalization rule with
F(t) = (t/(1 + t))
k
and φ(t) = (1 + t)
2
, and try to
find a true generating function. We have to solve the
equation:
w = tφ(w) or w = t(1 +w)
2
.
This equation is tw
2
− (1 − 2t)w + t = 0 and we are
looking for the unique solution w = w(t) such that
w(0) = 0. This is:
w(t) =
1 −2t −
√
1 −4t
2t
.
We now perform the necessary computations:
F(w) =

w
1 +w

k
=
62 CHAPTER 5. RIORDAN ARRAYS
=

1 −2t −
√
1 −4t
1 −
√
1 −4t

k
=
=

1 −
√
1 −4t
2

k
;
furthermore:
1
1 −tφ
′
(w)
=
1
1 −2t(1 +w)
=
1
√
1 −4t
.
Therefore, the diagonalization gives:

2n −k
n −k

= [t
n
]
1
√
1 −4t

1 −
√
1 −4t
2

k
.
This shows that the binomial coefficient is the generic
element of the Riordan array:
D =

1
√
1 −4t
,
1 −
√
1 −4t
2t

.
As a check, we observe that column 0 contains all the
elements with k = 0, i.e.,

2n
n

, and this is in accor-
dance with the generating function d(t) = 1/
√
1 −4t.
A simple example is:
n
¸
k=0

2n −k
n −k

2
k
=
= [t
n
]
1
√
1 −4t
¸
1
1 −2y


y =
1 −
√
1 −4t
2

=
= [t
n
]
1
√
1 −4t
1
√
1 −4t
= [t
n
]
1
1 −4t
= 4
n
.
By using the diagonalization rule as above, we can
show that:

2n +ak
n −ck

k∈N
=
=

1
√
1 −4t
, t
c−1

1 −
√
1 −4t
2t

a+2c

.
An interesting example is given by the following al-
ternating sum:
¸
k

2n
n −3k

(−1)
k
=
= [t
n
]
1
√
1 −4t
¸
1
1 +y


y = t
3

1 −
√
1 −4t
2t

6
¸
= [t
n
]

1
2
√
1 −4t
+
1 −t
2(1 −3t)

=
=
1
2

2n
n

+ 3
n−1
+
δ
n,0
6
.
The reader is invited to solve, in a similar way, the
corresponding non-alternating sum.
In the same way we can deal with binomial coeffi-
cients of the form

pn+ak
n−ck

, but in this case, in order
to apply the LIF, we have to solve an equation of de-
gree p > 2. This creates many difficulties, and we do
not insist on it any longer.
5.7 Coloured walks
In the section “Walks, trees and Catalan numbers”
we introduced the concept of a walk or path on the
integral lattice Z
2
. The concept can be generalized by
defining a walk as a sequence of steps starting from
the origin and composed by three kinds of steps:
1. east steps, which go from the point (x, y) to (x+
1, y);
2. diagonal steps, which go from the point (x, y) to
(x + 1, y + 1);
3. north steps, which go from the point (x, y) to
(x, y + 1).
A colored walk is a walk in which every kind of step
can assume different colors; we denote by a, b, c (a >
0, b, c ≥ 0) the number of colors the east, diagonal
and north steps can be. We discuss complete colored
walks, i.e., walks without any restriction, and under-
diagonal walks, i.e., walks that never go above the
main diagonal x −y = 0. The length of a walk is the
number of its steps, and we denote by d
n,k
the num-
ber of colored walks which have length n and reach a
distance k from the main diagonal, i.e., the last step
ends on the diagonal x − y = k ≥ 0. A colored walk
problem is any (counting) problem corresponding to
colored walks; a problem is called symmetric if and
only if a = c.
We wish to point out that our considerations are
by no means limited to the walks on the integral lat-
tice. Many combinatorial problems can be proved
to be equivalent to some walk problems; bracketing
problems are a typical example and, in fact, a vast
literature exists on walk problems.
Let us consider d
n+1,k+1
, i.e., the number of col-
ored walks of length n+1 reaching the distance k +1
from the main diagonal. We observe that each walk
is obtained in a unique way as:
1. a walk of length n reaching the distance k from
the main diagonal, followed by any of the a east
steps;
2. a walk of length n reaching the distance k + 1
from the main diagonal, followed by any of the b
diagonal steps;
3. a walk of length n reaching the distance k + 2
from the main diagonal, followed by any of the
c north steps.
Hence we have: d
n+1,k+1
= ad
n,k
+bd
n,k+1
+cd
n,k+2
.
This proves that A = ¦a, b, c¦ is the A-sequence of
(d
n,k
)
n,k∈N
, which therefore is a proper Riordan ar-
ray. This significant fact can be stated as:
5.7. COLOURED WALKS 63
Theorem 5.7.1 Let d
n,k
be the number of colored
walks of length n reaching a distance k from the main
diagonal, then the infinite triangle (d
n,k
)
n,k∈N
is a
proper Riordan array.
The Pascal, Catalan and Motzkin triangles define
walking problems that have different values of a, b, c.
When c = 0, it is easily proved that d
n,k
=

n
k

a
k
b
n−k
and so we end up with the Pascal triangle. Con-
sequently, we assume c = 0. For any given triple
(a, b, c) we obtain one type of array from complete
walks and another from underdiagonal walks. How-
ever, the function h(t), that only depends on the A-
sequence, is the same in both cases, and we can find it
by means of formula (5.3.2). In fact, A(t) = a+bt+ct
2
and h(t) is the solution of the functional equation
h(t) = a +bth(t) +ct
2
h(t)
2
having h(0) = 0:
h(t) =
1 −bt −
√
1 −2bt +b
2
t
2
−4act
2
2ct
2
(5.7.1)
The radicand 1 − 2bt + (b
2
− 4ac)t
2
= (1 − (b +
2
√
ac)t)(1 − (b − 2
√
ac)t) will be simply denoted by
∆.
Let us now focus our attention on underdiagonal
walks. If we consider d
n+1,0
, we observe that every
walk returning to the main diagonal can only be ob-
tained from another walk returning to the main diag-
onal followed by any diagonal step, or a walk ending
at distance 1 from the main diagonal followed by any
north step. Hence, we have d
n+1,0
= bd
n,0
+ cd
n,1
and in the column generating functions this corre-
sponds to d(t) − 1 = btd(t) + ctd(t)th(t). From this
relation we easily find d(t) = (1/a)h(t), and therefore
by (5.7.1) the Riordan array of underdiagonal colored
walk is:
(d
n,k
)
n,k∈N
=

1 −bt −
√
∆
2act
2
,
1 −bt −
√
∆
2ct
2

.
In current literature, major importance is usually
given to the following three quantities:
1. the number of walks returning to the main diag-
onal; this is d
n
= [t
n
]d(t), for every n,
2. the total number of walks of length n; this is
α
n
=
¸
n
k=0
d
n,k
, i.e., the value of the row sums
of the Riordan array;
3. the average distance from the main diagonal of
all the walks of length n; this is δ
n
=
¸
n
k=0
kd
n,k
,
which is the weighted row sum of the Riordan
array, divided by α
n
.
In Chapter 7 we will learn how to find an asymp-
totic approximation for d
n
. With regard to the last
two points, the formulas for the row sums and the
weighted row sums given in the first section allow us
to find the generating functions α(t) of the total num-
ber α
n
of underdiagonal walks of length n, and δ(t)
of the total distance δ
n
of these walks from the main
diagonal:
α(t) =
1
2at
1 −(b + 2a)t −
√
∆
(a +b +c)t −1
δ(t) =
1
4at

1 −(b + 2a)t −
√
∆
(a +b +c)t −1

2
.
In the symmetric case these formulas simplify as fol-
lows:
α(t) =
1
2at

1 −(b −2a)t
1 −(b + 2a)t
−1

δ(t) =
1
2at

1 −bt
1 −(b + 2a)t
−

1 −(b −2a)t
1 −(b + 2a)t

.
The alternating row sums and the diagonal sums
sometimes have some combinatorial significance as
well, and so they can be treated in the same way.
The study of complete walks follows the same lines
and we only have to derive the form of the corre-
sponding Riordan array, which is:
(d
n,k
)
n,k∈N
=

1
√
∆
,
1 −bt −
√
∆
2ct
2

.
The proof is as follows. Since a complete walk can
go above the main diagonal, the array (d
n,k
)
n,k∈N
is
only the right part of an infinite triangle, in which
k can also assume the negative values. By following
the logic of the theorem above, we see that the gen-
erating function of the nth row is ((c/w) +b +aw)
n
,
and therefore the bivariate generating function of the
extended triangle is:
d(t, w) =
¸
n

c
w
+b +aw

n
t
n
=
=
1
1 −(aw +b +c/w)t
.
If we expand this expression by partial fractions, we
get:
d(t, w) =
1
√
∆

1
1 −
1−bt−
√
∆
2ct
w
−
1
1 −
1−bt+
√
∆
2ct
w

=
1
√
∆

1
1 −
1−bt−
√
∆
2ct
w
+
+
1 −bt −
√
∆
2at
1
w
1
1 −
1−bt−
√
∆
2ct
1
w

.
64 CHAPTER 5. RIORDAN ARRAYS
The first term represents the right part of the ex-
tended triangle and this corresponds to k ≥ 0,
whereas the second term corresponds to the left part
(k < 0). We are interested in the right part, and the
expression can be written as:
1
√
∆
1
1 −
1−bt−
√
∆
2ct
w
=
1
√
∆
¸
k

1 −bt −
√
∆
2ct

k
w
k
which immediately gives the form of the Riordan ar-
ray.
5.8 Stirling numbers and Rior-
dan arrays
The connection between Riordan arrays and Stirling
numbers is not immediate. If we examine the two in-
finite triangles of the Stirling numbers of both kinds,
we immediately realize that they are not Riordan ar-
rays. It is not difficult to obtain the column generat-
ing functions for the Stirling numbers of the second
kind; by starting with the recurrence relation:

n + 1
k + 1

= (k + 1)

n
k + 1

+

n
k
¸
and the obvious generating function S
0
(t) = 1, we
can specialize the recurrence (valid for every n ∈ N)
to the case k = 0. This gives the relation between
generating functions:
S
1
(t) −S
1
(0)
t
= S
1
(t) +S
0
(t);
because S
1
(0) = 0, we immediately obtain S
1
(t) =
t/(1−t), which is easily checked by looking at column
1 in the array. In a similar way, by specializing the
recurrence relation to k = 1, we find S
2
(t) = 2tS
2
(t)+
tS
1
(t), whose solution is:
S
2
(t) =
t
2
(1 −t)(1 −2t)
.
This proves, in an algebraic way, that
¸
n
2
¸
= 2
n−1
−1,
and also indicates the form of the generating function
for column m:
S
m
(t) = (

n
m
¸
n∈N
=
t
m
(1 −t)(1 −2t) (1 −mt)
which is now proved by induction when we specialize
the recurrence relation above to k = m. This is left
to the reader as a simple exercise.
The generating functions for the Stirling numbers
of the first kind are not so simple. However, let us
go on with the Stirling numbers of the second kind
proceeding in the following way; if we multiply the
recurrence relation by (k +1)!/(n+1)! we obtain the
new relation:
(k + 1)!
(n + 1)!

n + 1
k + 1

=
=
(k + 1)!
n!

n
k + 1

k + 1
n + 1
+
k!
n!

n
k
¸
k + 1
n + 1
.
If we denote by d
n,k
the quantity k!
¸
n
k
¸
/n!, this is a
recurrence relation for d
n,k
, which can be written as:
(n + 1)d
n+1,k+1
= (k + 1)d
n,k+1
+ (k + 1)d
n,k
.
Let us now proceed as above and find the column
generating functions for the new array (d
n,k
)
n,k∈N
.
Obviously, d
0
(t) = 1; by setting k = 0 in the new
recurrence:
(n + 1)d
n+1,1
= d
n,1
+d
n,0
and passing to generating functions: d
′
1
(t) = d
1
(t) +
1. The solution of this simple differential equation
is d
1
(t) = e
t
− 1 (the reader can simply check this
solution, if he or she prefers). We can now go on
by setting k = 1 in the recurrence; we obtain: (n +
1)d
n+1,2
= 2d
n,2
+ 2d
n,1
, or d
′
2
(t) = 2d
2
(t) + 2(e
t
−
1). Again, this differential equation has the solution
d
2
(t) = (e
t
− 1)
2
, and this suggests that, in general,
we have: d
k
(t) = (e
t
− 1)
k
. A rigorous proof of this
fact can be obtained by mathematical induction; the
recurrence relation gives: d
′
k+1
(t) = (k +1)d
k+1
(t) +
(k + 1)d
k
(t). By the induction hypothesis, we can
substitute d
k
(t) = (e
t
−1)
k
and solve the differential
equation thus obtained. In practice, we can simply
verify that d
k+1
(t) = (e
t
−1)
k+1
; by substituting, we
have:
(k + 1)e
t
(e
t
−1)
k
=
(k + 1)(e
t
−1)
k+1
+ (k + 1)(e
t
−1)
k
and this equality is obviously true.
The form of this generating function:
d
k
(t) = (

k!
n!

n
k
¸

n∈N
= (e
t
−1)
k
proves that (d
n,k
)
n,k∈N
is a Riordan array having
d(t) = 1 and th(t) = (e
t
− 1). This fact allows us
to prove algebraically a lot of identities concerning
the Stirling numbers of the second kind, as we shall
see in the next section.
For the Stirling numbers of the first kind we pro-
ceed in an analogous way. We multiply the basic
recurrence:
¸
n + 1
k + 1

= n
¸
n
k + 1

+

n
k

5.9. IDENTITIES INVOLVING THE STIRLING NUMBERS 65
by (k + 1)!/(n + 1)! and study the quantity f
n,k
=
k!

n
k

/n!:
(k + 1)!
(n + 1)!
¸
n + 1
k + 1

=
=
(k + 1)!
n!
¸
n
k + 1

n
n + 1
+
k!
n!

n
k

k + 1
n + 1
,
that is:
(n + 1)f
n+1,k+1
= nf
n,k+1
+ (k + 1)f
n,k
.
In this case also we have f
0
(t) = 1 and by special-
izing the last relation to the case k = 0, we obtain:
f
′
1
(t) = tf
′
1
(t) +f
0
(t).
This is equivalent to f
′
1
(t) = 1/(1 − t) and because
f
1
(0) = 0 we have:
f
1
(t) = ln
1
1 −t
.
By setting k = 1, we find the simple differential equa-
tion f
′
2
(t) = tf
′
2
(t) + 2f
1
(t), whose solution is:
f
2
(t) =

ln
1
1 −t

2
.
This suggests the general formula:
f
k
(t) = (

k!
n!

n
k


n∈N
=

ln
1
1 −t

k
and again this can be proved by induction. In this
case, (f
n,k
)
n,k∈N
is the Riordan array having d(t) = 1
and th(t) = ln(1/(1 −t)).
5.9 Identities involving the
Stirling numbers
The two recurrence relations for d
n,k
and f
n,k
do not
give an immediate evidence that the two triangles
are indeed Riordan arrays., because they do not cor-
respond to A-sequences. However, the A-sequences
for the two arrays can be easily found, once we know
their h(t) function. For the Stirling numbers of the
first kind we have to solve the functional equation:
ln
1
1 −t
= tA

ln
1
1 −t

.
By setting y = ln(1/(1 − t)) or t = (e
y
− 1)/y, we
have A(y) = ye
y
/(e
y
− 1) and this is the generating
function for the A-sequence we were looking for. In
a similar way, we find that the A-sequence for the
triangle related to the Stirling numbers of the second
kind is:
A(t) =
t
ln(1 +t)
.
A first result we obtain by using the correspon-
dence between Stirling numbers and Riordan arrays
concerns the row sums of the two triangles. For the
Stirling numbers of the first kind we have:
n
¸
k=0

n
k

= n!
n
¸
k=0
k!
n!

n
k

1
k!
=
= n![t
n
]
¸
e
y


y = ln
1
1 −t

=
= n![t
n
]
1
1 −t
= n!
as we observed and proved in a combinatorial way.
The row sums of the Stirling numbers of the second
kind give, as we know, the Bell numbers; thus we
can obtain the (exponential) generating function for
these numbers:
n
¸
k=0

n
k
¸
= n!
n
¸
k=0
k!
n!

n
k
¸
1
k!
=
= n![t
n
]

e
y


y = e
t
−1

=
= n![t
n
] exp(e
t
−1);
therefore we have:
(

B
n
n!

= exp(e
t
−1).
We also defined the ordered Bell numbers as O
n
=
¸
n
k=0
¸
n
k
¸
k!; therefore we have:
O
n
n!
=
n
¸
k=0
k!
n!

n
k
¸
=
= [t
n
]
¸
1
1 −y


y = e
t
−1

= [t
n
]
1
2 −e
t
.
We have thus obtained the exponential generating
function:
(

O
n
n!

=
1
2 −e
t
.
Stirling numbers of the two kinds are related be-
tween them in various ways. For example, we have:
¸
k

n
k


k
m

=
n!
m!
¸
k
k!
n!

n
k

m!
k!

k
m

=
=
n!
m!
[t
n
]
¸
(e
y
−1)
m


y = ln
1
1 −t

=
=
n!
m!
[t
n
]
t
m
(1 −t)
m
=
n!
m!

n −1
m−1

.
Besides, two orthogonality relations exist between
Stirling numbers. The first one is proved in this way:
¸
k

n
k


k
m

(−1)
n−k
=
66 CHAPTER 5. RIORDAN ARRAYS
= (−1)
n
n!
m!
¸
k
k!
n!

n
k

m!
k!

k
m

(−1)
k
=
= (−1)
n
n!
m!
[t
n
]
¸
(e
−y
−1)
m


y = ln
1
1 −t

=
= (−1)
n
n!
m!
[t
n
](−t)
m
= δ
n,m
.
The second orthogonality relation is proved in a sim-
ilar way and reads:
¸
k

n
k
¸
¸
k
m

(−1)
n−k
= δ
n,m
.
We introduced Stirling numbers by means of Stir-
ling identities relative to powers and falling factorials.
We can now prove these identities by using a Riordan
array approach. In fact:
n
¸
k=0

n
k

(−1)
n−k
x
k
=
= (−1)
n
n!
n
¸
k=0
k!
n!

n
k

x
k
k!
(−1)
k
=
= (−1)
n
n![t
n
]
¸
e
−xy


y = ln
1
1 −t

=
= (−1)
n
n![t
n
](1 −t)
x
= n!

x
n

= x
n
and:
n
¸
k=0

n
k
¸
x
k
= n!
n
¸
k=0
k!
n!

n
k
¸

x
k

=
= n![t
n
]

(1 +y)
x


y = e
t
−1

=
= n![t
n
]e
tx
= x
n
.
We conclude this section by showing two possible
connections between Stirling numbers and Bernoulli
numbers. First we have:
n
¸
k=0

n
k
¸
(−1)
k
k!
k + 1
= n!
n
¸
k=0
k!
n!

n
k
¸
(−1)
k
k + 1
=
= n![t
n
]
¸
−
1
y
ln
1
1 +y


y = e
t
−1

=
= n![t
n
]
t
e
t
−1
= B
n
which proves that Bernoulli numbers can be defined
in terms of the Stirling numbers of the second kind.
For the Stirling numbers of the first kind we have the
identity:
n
¸
k=0

n
k

B
k
= n!
n
¸
k=0
k!
n!

n
k

B
k
k!
=
= n![t
n
]
¸
y
e
y
−1


y = ln
1
1 −t

=
= n![t
n
]
1 −t
t
ln
1
1 −t
=
= n![t
n+1
](1 −t) ln
1
1 −t
=
(n −1)!
n + 1
.
Clearly, this holds for n > 0. For n = 0 we have:
n
¸
k=0

n
k

B
k
= B
0
= 1.
Chapter 6
Formal methods
6.1 Formal languages
During the 1950’s, the linguist Noam Chomski in-
troduced the concept of a formal language. Several
definitions have to be provided before a precise state-
ment of the concept can be given. Therefore, let us
proceed in the following way.
First, we recall definitions given in Section 2.1. An
alphabet is a finite set A = ¦a
1
, a
2
, . . . , a
n
¦, whose
elements are called symbols or letters. A word on A
is a finite sequence of symbols in A; the sequence is
written by juxtaposing the symbols, and therefore a
word w is denoted by w = a
i1
a
i2
. . . a
ir
, and r = [w[ is
the length of the word. The empty sequence is called
the empty word and is conventionally denoted by ǫ;
its length is obviously 0, and is the only word of 0
length.
The set of all the words on A, the empty word in-
cluded, is indicated by A
∗
, and by A
+
if the empty
word is excluded. Algebraically, A
∗
is the free monoid
on A, that is the monoid freely generated by the sym-
bols in A. To understand this point, let us consider
the operation of juxtaposition and recursively apply
it starting with the symbols in A. What we get are
the words on A, and the juxtaposition can be seen as
an operation between them. The algebraic structure
thus obtained has the following properties:
1. associativity: w
1
(w
2
w
3
) = (w
1
w
2
)w
3
=
w
1
w
2
w
3
;
2. ǫ is the identity or neutral element: ǫw = wǫ =
w.
It is called a monoid, which, by construction, has
been generated by combining the symbols in A in all
the possible ways. Because of that, (A
∗
, ), if de-
notes the juxtaposition, is called the “free monoid”
generated by A. Observe that a monoid is an alge-
braic structure more general than a group, in which
all the elements have an inverse as well.
If w ∈ A
∗
and z is a word such that w can be
decomposed w = w
1
zw
2
(w
1
and/or w
2
possibly
empty), we say that z is a subword of w; we also
say that z occurs in w, and the particular instance
of z in w is called an occurrence of z in w. Observe
that if z is a subword of w, it can have more than one
occurrence in w. If w = zw
2
, we say that z is a head
or prefix of w, and if w = w
1
z, we say that z is a tail
or suffix of w. Finally, a language on A is any subset
L ⊆ A
∗
.
The basic definition concerning formal languages
is the following: a grammar is a 4-tuple G =
(T, N, σ, {), where:
• T = ¦a
1
, a
2
, . . . , a
n
¦ is the alphabet of terminal
symbols;
• N = ¦φ
1
, φ
2
, . . . , φ
m
¦ is the alphabet of non-
terminal symbols;
• σ ∈ N is the initial symbol;
• { is a finite set of productions.
Usually, the symbols in T are denoted by lower
case Latin letters; the symbols in N by Greek let-
ters or by upper case Latin letters. A production
is a pair (z
1
, z
2
) of words in T ∪ N, such that z
1
contains at least a symbol in N; the production is
often indicated by z
1
→ z
2
. If w ∈ (T ∪ N)
∗
, we
can apply a production z
1
→ z
2
∈ { to w whenever
w can be decomposed w = w
1
z
1
w
2
, and the result
is the new word w
1
z
2
w
2
∈ (T ∪ N)
∗
; we will write
w = w
1
z
1
w
2
⊢ w
1
z
2
w
2
when w
1
z
1
w
2
is the decompo-
sition of w in which z
1
is the leftmost occurrence of
z
1
in w; in other words, if we also have w = ´ w
1
z
1
´ w
2
,
then [w
1
[ < [ ´ w
1
[.
Given a grammar G = (T, N, σ, {), we define the
relation w ⊢ ´ w between words w, ´ w ∈ (T ∪ N)
∗
: the
relation holds if and only if a production z
1
→ z
2
∈ {
exists such that z
1
occurs in w, w = w
1
z
1
w
2
is the
leftmost occurrence of z
1
in w and ´ w = w
1
z
2
w
2
. We
also denote by ⊢
∗
the transitive closure of ⊢ and call
it generation or derivation; this means that w ⊢
∗
´ w
if and only if a sequence (w = w
1
, w
2
, . . . , w
s
= ´ w)
exists such that w
1
⊢ w
2
, w
2
⊢ w
3
, . . . , w
s−1
⊢ w
s
.
We observe explicitly that by our condition that in
every production z
1
→ z
2
the word z
1
should contain
67
68 CHAPTER 6. FORMAL METHODS
at least a symbol in N, if a word w
i
∈ T
∗
is produced
during a generation, it is terminal, i.e., the generation
should stop. By collecting all these definitions, we
finally define the language generated by the grammar
G as the set:
L(G) =

w ∈ T
∗


σ ⊢
∗
w
¸
i.e., a word w ∈ T
∗
is in L(G) if and only if we can
generate it by starting with the initial symbol σ and
go on by applying the productions in { until w is
generated. At that moment, the generation stops.
Note that, sometimes, the generation can go on for-
ever, never generating a word on T; however, this is
not a problem: it only means that such generations
should be ignored.
6.2 Context-free languages
The definition of a formal language is quite general
and it is possible to show that formal languages co-
incide with the class of “partially recursive sets”, the
largest class of sets which can be constructed recur-
sively, i.e., in finite terms. This means that we can
give rules to build such sets (e.g., we can give a gram-
mar for them), but their construction can go on for-
ever, so that, looking at them from another point of
view, if we wish to know whether a word w belongs
to such a set S, we can be unlucky and an infinite
process can be necessary to find out that w ∈ S.
Because of that, people have studied more re-
stricted classes of languages, for which a finite process
is always possible for finding out whether w belongs
to the language or not. Surely, the most important
class of this kind is the class of “context-free lan-
guages”. They are defined in the following way. A
context-free grammar is a grammar G = (T, N, σ, {)
in which all the productions z
1
→ z
2
in { are such
that z
1
∈ N. The naming “context-free” derives from
this definition, because a production z
1
→ z
2
is ap-
plied whenever the non-terminal symbol z
1
is the left-
most non-terminal symbol in a word, irrespective of
the context in which it appears.
As a very simple example, let us consider the fol-
lowing grammar. Let T = ¦a, b¦, N = ¦σ¦; σ is
the initial symbol of the grammar, being the only
non-terminal. The set { is composed of the two pro-
ductions:
σ → ǫ σ → aσbσ.
This grammar is called the Dyck grammar and the
language generated by it the Dyck language. In Fig-
ure 6.1 we draw the generation of some words in the
Dyck language. The recursive nature of the produc-
tions allows us to prove properties of the Dyck lan-
guage by means of mathematical induction:
Theorem 6.2.1 A word w ∈ ¦a, b¦
∗
belongs to the
Dyck language D if and only if:
i) the number of a’s in w equals the number of b’s;
ii) in every prefix z of w the number of a’s is not
less than the number of b’s.
Proof: Let w ∈ D; if w = ǫ nothing has to be
proved. Otherwise, w is generated by the second pro-
duction and w = aw
1
bw
2
with w
1
, w
2
∈ D; therefore,
if we suppose that i) holds for w
1
and w
2
, it also
holds for w. For ii), any prefix z of w must have
one of the forms: a, az
1
where z
1
is a prefix of w
1
,
aw
1
b or aw
1
bz
2
where z
2
is a prefix of w
2
. By the
induction hypothesis, ii) should hold for z
1
and z
2
,
and therefore it is easily proved for w. Vice versa, let
us suppose that i) and ii) hold for w ∈ T
∗
. If w = ǫ,
then by ii) w should begin by a. Let us scan w until
we find the first occurrence of the symbol b such that
w = aw
1
bw
2
and in w
1
the number of b’s equals the
number of a’s. By i) such occurrence of b must exist,
and consequently w
1
and w
2
must satisfy condition
i). Besides, if w
1
and w
2
are not empty, then they
should satisfy condition ii), by the very construction
of w
1
and the fact that w satisfies condition ii) by
hypothesis. We have thus obtained a decomposition
of w showing that the second production has been
used. This completes the proof.
If we substitute the letter a with the symbol ‘(
′
and
the letter b with the symbol ‘)’, the theorem shows
that the words in the Dyck language are the possi-
ble parenthetizations of an expression. Therefore, the
number of Dyck words with n pairs of parentheses is
the Catalan number

2n
n

/(n+1). We will see how this
result can also be obtained by starting with the def-
inition of the Dyck language and applying a suitable
and mechanical method, known as Sch¨ utzenberger
methodology or symbolic method. The method can
be applied to every set of objects, which are defined
through a non-ambiguous context-free grammar.
A context-free grammar G is ambiguous iff there
exists a word w ∈ L(G) which can be generated by
two different leftmost derivations. In other words, a
context-free grammar H is non-ambiguous iff every
word w ∈ L(H) can be generated in one and only
one way. An example of an ambiguous grammar is
G = (T, N, σ, {) where T = ¦1¦, N = ¦σ¦ and {
contains the two productions:
σ → 1 σ → σσ.
For example, the word 111 is generated by the two
following leftmost derivations:
σ ⊢ σσ ⊢ 1σ ⊢ 1σσ ⊢ 11σ ⊢ 111
σ ⊢ σσ ⊢ σσσ ⊢ 1σσ ⊢ 11σ ⊢ 111.
6.3. FORMAL LANGUAGES AND PROGRAMMING LANGUAGES 69
σ
ǫ
aσbσ
abσ aaσbσbσ
ab abaσbσ aabσbσ aaaσbσbσbσ
ababσ abaaσbσbσ aabbσ aabaσbσbσ
abab ababaσbσ aabb aabbaσbσ
$
$
$
$
$
$
$ ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
@
@
@
@
@
@
@
@
@
@@ h
h
h
h
h
h
h
h
h
hh

d
d
d
d

d
d
d
d
d
d

d
d
d
d
d
d
d
d
Figure 6.1: The generation of some Dyck words
Instead, the Dyck grammar is non-ambiguous; in
fact, as we have shown in the proof of the pre-
vious theorem, given any word w ∈ D, w = ǫ,
there is only one decomposition w = aw
1
bw
2
, hav-
ing w
1
, w
2
∈ D; therefore, w can only be generated
in a single way. In general, if we show that any word
in a context-free language L(G), generated by some
grammar G, has a unique decomposition according
to the productions in G, then the grammar cannot
be ambiguous. Because of the connection between
the Sch¨ utzenberger methodology and non-ambiguous
context-free grammars, we are mainly interested in
this kind of grammars. For the sake of completeness,
a context-free language is called intrinsically ambigu-
ous iff every context-free grammar generating it is
ambiguous. This definition stresses the fact that, if a
language is generated by an ambiguous grammar, it
can also be generated by some non-ambiguous gram-
mar, unless it is intrinsically ambiguous. It is possible
to show that intrinsically ambiguous languages actu-
ally exist; fortunately, they are not very frequent. For
example, the language generated by the previous am-
biguous grammar is ¦1¦
+
, i.e., the set of all the words
composed by any sequence of 1’s, except the empty
word. actually, it is not an ambiguous language and
a non-ambiguous grammar generating it is given by
the same T, N, σ and the two productions:
σ → 1 σ → 1σ.
It is a simple matter to show that every word 11 . . . 1
can be uniquely decomposed according to these pro-
ductions.
6.3 Formal languages and pro-
gramming languages
In 1960, the formal definition of the programming
language ALGOL’60 was published. ALGOL’60 has
surely been the most influential programming lan-
guage ever created, although it was actually used only
by a very limited number of programmers. Most of
the concepts we now find in programming languages
were introduced by ALGOL’60, of which, for exam-
ple, PASCAL and C are direct derivations. Here,
we are not interested in these aspects of ALGOL’60,
but we wish to spend some words on how ALGOL’60
used context-free grammars to define its syntax in a
formal and precise way. In practice, a program in
ALGOL’60 is a word generated by a (rather com-
plex) context-free grammar, whose initial symbol is
'program`.
The ALGOL’60 grammar used, as terminal sym-
bol alphabet, the characters available on the stan-
dard keyboard of a computer; actually, they were the
characters punchable on a card, the input mean used
at that time to introduce a program into the com-
puter. The non-terminal symbol notation was one
of the most appealing inventions of ALGOL’60: the
symbols were composed by entire English sentences
enclosed by the two special parentheses ' and `. This
allowed to clearly express the intended meaning of the
non-terminal symbols. The previous example con-
cerning 'program` makes surely sense. Another tech-
nical device used by ALGOL’60 was the compaction
of productions; if we had several production with the
same left hand symbol β → w
1
, β → w
2
, . . . , β → w
k
,
70 CHAPTER 6. FORMAL METHODS
they were written as a single rule:
β ::= w
1
[ w
2
[ [ w
k
where ::= was a metasymbol denoting definition and
[ was read “or” to denote alternatives. This notation
is usually called Backus Normal Form (BNF).
Just to do a very simple example, in Figure 6.1
(lines 1 through 6) we show how integer numbers
were defined. This definition avoids leading 0’s in
numbers, but allows both +0 and −0. Productions
can be easily changed to avoid +0 or −0 or both.
In the same figure, line 7 shows the definition of the
conditional statements.
This kind of definition gives a precise formula-
tion of all the clauses in the programming language.
Besides, since the program has a single generation
according to the grammar, it is possible to find
this derivation starting from the actual program and
therefore give its exact structure. This allows to give
precise information to the compiler, which, in a sense,
is directed from the formal syntax of the language
(syntax directed compilation).
A very interesting aspect is how this context-free
grammar definition can avoid ambiguities in the in-
terpretation of a program. Let us consider an expres-
sion like a + b ∗ c; according to the rules of Algebra,
the multiplication should be executed before the ad-
dition, and the computer must follow this convention
in order to create no confusion. This is done by the
simplified productions given by lines 8 through 11 in
Figure 6.1. The derivation of the simple expression
a+b ∗ c, or of a more complicated expression, reveals
that it is decomposed into the sum of a and b ∗ c;
this information is passed to the compiler and the
multiplication is actually performed before addition.
If powers are also present, they are executed before
products.
This ability of context-free grammars in design-
ing the syntax of programming languages is very im-
portant, and after ALGOL’60 the syntax of every
programming language has always been defined by
context-free grammars. We conclude by remembering
that a more sophisticated approach to the definition
of programming languages was tried with ALGOL’68
by means of van Wijngaarden’s grammars, but the
method revealed too complex and was abandoned.
6.4 The symbolic method
The Sch¨ utzenberger’s method allows us to obtain
the counting generating function for every non-
ambiguous language, starting with the correspond-
ing non-ambiguous grammar and proceeding in a me-
chanical way. Let us begin by a simple example; Fi-
bonacci words are the words on the alphabet ¦0, 1¦
beginning and ending by the symbol 1 and never con-
taining two consecutive 0’s. For small values of n,
Fibonacci words of length n are easily displayed:
n = 1 1
n = 2 11
n = 3 111, 101
n = 4 1111, 1011, 1101
n = 5 11111, 10111, 11011, 11101, 10101
If we count them by their length, we obtain the
sequence ¦0, 1, 1, 2, 3, 5, 8, . . .¦, which is easily recog-
nized as the Fibonacci sequence. In fact, a word of
length n is obtained by adding a trailing 1 to a word of
length n−1, or adding a trailing 01 to a word of length
n − 2. This immediately shows, in a combinatorial
way, that Fibonacci words are counted by Fibonacci
numbers. Besides, we get the productions of a non-
ambiguous context-free grammar G = (T, N, σ, {),
where T = ¦0, 1¦, N = ¦φ¦, σ = φ and { contains:
φ → 1 φ → φ1 φ → φ01
(these productions could have been written φ ::=
1 [ φ1 [ φ01 by using the ALGOL’60 notations).
We are now going to obtain the counting gener-
ating function for Fibonacci words by applying the
Sch¨ utzenberger’s method. This consists in the fol-
lowing steps:
1. every non-terminal symbol σ ∈ N is transformed
into the name of its counting generating function
σ(t);
2. every terminal symbol is transformed into t;
3. the empty word is transformed into 1;
4. every [ sign is transformed into a + sign, and ::=
is transformed into an equal sign.
After having performed these transformations, we ob-
tain a system of equations, which can be solved in the
unknown generating functions introduced in the first
step. They are the counting generating functions for
the languages generated by the corresponding non-
terminal symbols, when we consider them as the ini-
tial symbols.
The definition of the Fibonacci words produces:
φ(t) = t +tφ(t) +t
2
φ(t)
the solution of which is:
φ(t) =
t
1 −t −t
2
;
this is obviously the generating function for the Fi-
bonacci numbers. Therefore, we have shown that the
6.5. THE BIVARIATE CASE 71
1 'digit` ::= 0 [ 1 [ 2 [ 3 [ 4 [ 5 [ 6 [ 7 [ 8 [ 9
2 'non −zero digit` ::= 1 [ 2 [ 3 [ 4 [ 5 [ 6 [ 7 [ 8 [ 9
3 'sequence of digits` ::= 'digit` [ 'digit` 'sequence of digits`
4 'unsigned number` ::= 'digit` [ 'non −zero digit` 'sequence of digits`
5 'signed number` ::= +'unsigned number` [ −'unsigned number`
6 'integer number` ::= 'unsigned number` [ 'signed number`
7 'conditional clause` ::= if 'condition` then 'instruction` [
if 'condition` then 'instruction` else 'instruction`
8 'expression` ::= 'term` [ 'term` +'expression`
9 'term` ::= 'factor` [ 'term` ∗ 'factor`
10 'factor` ::= 'element` [ 'factor` ↑ 'element`
11 'element` ::= 'constant` [ 'variable` [ ('expression`)
Table 6.1: Context-free languages and programming languages
number of Fibonacci words of length n is F
n
, as we
have already proved by combinatorial arguments.
In the case of the Dyck language, the definition
yields:
σ(t) = 1 +t
2
σ(t)
2
and therefore:
σ(t) =
1 −
√
1 −4t
2
2t
2
.
Since every word in the Dyck language has an even
length, the number of Dyck words with 2n symbols is
just the nth Catalan number, and this also we knew
by combinatorial means.
Another example is given by the Motzkin words;
these are words on the alphabet ¦a, b, c¦ in which
a, b act as parentheses in the Dyck language, while
c is free and can appear everywhere. Therefore, the
definition of the language is:
µ ::= ǫ [ cµ[ aµbµ
if µ is the only non-terminal symbol. The
Sch¨ utzenberger’s method gives the equation:
µ(t) = 1 +tµ(t) +t
2
µ(t)
2
whose solution is easily found:
µ(t) =
1 −t −
√
1 −2t −3t
2
2t
2
.
By expanding this function we find the sequence of
Motzkin numbers, beginning:
n 0 1 2 3 4 5 6 7 8 9
M
n
1 1 2 4 9 21 51 127 323 835
These numbers count the so-called unary-binary
trees, i.e., trees the nodes of which have ariety 1 or 2.
They can be defined in a pictorial way by means of
an object grammar:
An object grammar defines combinatorial objects
instead of simple letters or words; however, most
times, it is rather easy to pass from an object gram-
mar to an equivalent context-free grammar, and
therefore obtain counting generating functions by
means of Sch¨ utzenberger’s method. For example, the
object grammar in Figure 6.2 is obviously equivalent
to the context-free grammar for Motzkin words.
6.5 The bivariate case
In Sch¨ utzenberger’s method, the rˆole of the indeter-
minate t is to count the number of letters or symbols
occurring in the generated words; because of that, ev-
ery symbol appearing in a production is transformed
into a t. However, we can wish to count other param-
eters instead of or in conjunction with the number of
symbols. This is accomplished by modifying the in-
tended meaning of the indeterminate t and/or intro-
ducing some other indeterminate to take into account
the other parameters.
For example, in the Dyck language, we can wish to
count the number of pairs a, b occurring in the words;
this means that t no longer counts the single letters,
but counts the pairs. Therefore, the Sch¨ utzenberger’s
method gives the equation σ(t) = 1 + tσ(t)
2
, whose
solution is just the generating function of the Catalan
numbers.
An interesting application is as follows. Let us sup-
pose we wish to know how many Fibonacci words of
length n contain k zeroes. Besides the indeterminate
t counting the total number of symbols, we introduce
a new indeterminate z counting the number of zeroes.
From the productions of the Fibonacci grammar, we
derive an equation for the bivariate generating func-
tion φ(t, z), in which the coefficient of t
n
z
k
is just the
72 CHAPTER 6. FORMAL METHODS




t
t
t
t
µ
::=
`

ǫ
,




t
t
t
t
µ
,




d
d
d
d




t
t
t
t
µ




t
t
t
t
µ
Figure 6.2: An object grammar
number we are looking for. The equation is:
φ(t, z) = t +tφ(t, z) +t
2
zφ(t, z).
In fact, the production φ → φ1 increases by 1 the
length of the words, but does not introduce any 0;
the production φ → φ01, increases by 2 the length
and introduces a zero. By solving this equation, we
find:
φ(t, z) =
t
1 −t −zt
2
.
We can now extract the coefficient φ
n,k
of t
n
z
k
in the
following way:
φ
n,k
= [t
n
][z
k
]
t
1 −t −zt
2
=
= [t
n
][z
k
]
t
1 −t
1
1 −z
t
2
1−t
=
= [t
n
][z
k
]
t
1 −t
∞
¸
k=0
z
k

t
2
1 −t

k
=
= [t
n
][z
k
]
∞
¸
k=0
t
2k+1
z
k
(1 −t)
k+1
= [t
n
]
t
2k+1
(1 −t)
k+1
=
= [t
n−2k−1
]
1
(1 −t)
k+1
=

n −k −1
k

.
Therefore, the number of Fibonacci words of length
n containing k zeroes is counted by a binomial coeffi-
cient. The second expression in the derivation shows
that the array (φ
n,k
)
n,k∈N
is indeed a Riordan ar-
ray (t/(1 −t), t/(1 −t)), which is the Pascal triangle
stretched vertically, i.e., column k is shifted down by
k positions (k + 1, in reality). The general formula
we know for the row sums of a Riordan array gives:
¸
k
φ
n,k
=
¸
k

n −k −1
k

=
= [t
n
]
t
1 −t
¸
1
1 −y


y =
t
2
1 −t

=
= [t
n
]
t
1 −t −t
2
= F
n
as we were expecting. A more interesting problem
is to find the average number of zeroes in all the Fi-
bonacci words with n letters. First, we count the
total number of zeroes in all the words of length n:
¸
k
kφ
n,k
=
¸
k

n −k −1
k

k =
= [t
n
]
t
1 −t
¸
y
(1 −y)
2


y =
t
2
1 −t

=
= [t
n
]
t
3
(1 −t −t
2
)
2
.
We extract the coefficient:
[t
n
]
t
3
(1 −t −t
2
)
2
=
= [t
n−1
]

1
√
5

1
1 −φt
−
1
1 −
´
φt

2
=
=
1
5
[t
n−1
]
1
(1 −φt)
2
−
2
5
[t
n
]
t
(1 −φt)(1 −
´
φt)
+
+
1
5
[t
n−1
]
1
(1 −
´
φt)
2
=
=
1
5
[t
n−1
]
1
(1 −φt)
2
−
2
5
√
5
[t
n
]
1
1 −φt
+
+
2
5
√
5
[t
n
]
1
1 −
´
φt
+
1
5
[t
n−1
]
1
(1 −
´
φt)
2
.
The last two terms are negligible because they rapidly
tend to 0; therefore we have:
¸
k
kφ
n,k
≈
n
5
φ
n−1
−
2
5
√
5
φ
n
.
To obtain the average number Z
n
of zeroes, we need
to divide this quantity by F
n
∼ φ
n
/
√
5, the total
number of Fibonacci words of length n:
Z
n
=
¸
k
kφ
n,k
F
n
∼
n
φ
√
5
−
2
5
=
5 −
√
5
10
n −
2
5
.
This shows that the average number of zeroes grows
linearly with the length of the words and tends to
become the 27.64% of this length, because (5 −
√
5)/10 ≈ 0.2763932022 . . ..
6.6 The Shift Operator
In the usual mathematical terminology, an operator
is a mapping from some set T
1
of functions into some
6.7. THE DIFFERENCE OPERATOR 73
other set of functions T
2
. We have already encoun-
tered the operator (, acting from the set of sequences
(which are properly functions from N into R or C)
into the set of formal power series (analytic func-
tions). Other usual examples are D, the operator
of differentiation, and

, the operator of indefinite
integration. Since the middle of the 19th century,
the English mathematicians (G. Boole in particular)
introduced the concept of a finite operator, i.e., an
operator acting in finite terms, in contrast to an in-
finitesimal operator, such as differentiation and inte-
gration. The most simple among the finite operators
is the identity, denoted by I or by 1, which does not
change anything. Operationally, however, the most
important finite operator is the shift operator, de-
noted by E, changing the value of any function f at
a point x into the value of the same function at the
point x + 1. We write:
Ef(x) = f(x + 1)
Unlike an infinitesimal operator as D, a finite op-
erator can be applied to a sequence, as well, and in
that case we can write:
Ef
n
= f
n+1
This property is particularly interesting from our
point of view but, in order to follow the usual nota-
tional conventions, we shall adopt the first, functional
notation rather than the second one, more specific for
sequences.
The shift operator can be iterated and, to denote
the successive applications of E, we write E
n
instead
of EE . . . E (n times). So:
E
2
f(x) = EEf(x) = Ef(x + 1) = f(x + 2)
and in general:
E
n
f(x) = f(x +n)
Conventionally, we set E
0
= I = 1, and this is in
accordance with the meaning of I:
E
0
f(x) = f(x) = If(x)
We wish to observe that every recurrence can be
written using the shift operator. For example, the
recurrence for the Fibonacci numbers is:
E
2
F
n
= EF
n
+F
n
and this can be written in the following way, separat-
ing the operator parts from the sequence parts:
(E
2
−E −I)F
n
= 0
Some obvious properties of the shift operator are:
E(αf(x) +βg(x)) = αEf(x) +βEg(x)
E(f(x)g(x)) = Ef(x)Eg(x)
Hence, if c is any constant (i.e., c ∈ R or c ∈ C) we
have Ec = c.
It is possible to consider negative powers of E, as
well. So we have:
E
−1
f(x) = f(x −1) E
−n
f(x) = f(x −n)
and this is in accordance with the usual rules of pow-
ers:
E
n
E
m
= E
n+m
for n, m ∈ Z
E
n
E
−n
= E
0
= I for n ∈ Z
Finite operators are commonly used in Numerical
Analysis. In that case, an increment h is defined and
the shift operator acts according to this increment,
i.e., Ef(x) = f(x +h). When considering sequences,
this makes no sense and we constantly use h = 1.
We can have occasions to use two or more shift
operators, that is shift operators related to different
variables. We’ll distinguish them by suitable sub-
scripts:
E
x
f(x, y) = f(x + 1, y) E
y
f(x, y) = f(x, y + 1)
E
x
E
y
f(x, y) = E
y
E
x
f(x, y) = f(x + 1, y + 1)
6.7 The Difference Operator
The second most important finite operator is the dif-
ference operator ∆; it is defined in terms of the shift
and identity operators:
∆f(x) = (E −1)f(x) =
= Ef(x) −If(x) = f(x + 1) −f(x)
Some simple examples are:
∆c = Ec −c = c −c = 0 ∀c ∈ C
∆x = Ex −x = x + 1 −x = 1
∆x
2
= Ex
2
−x
2
= (x + 1)
2
−x
2
= 2x + 1
∆x
n
= Ex
n
−x
n
= (x + 1)
n
−x
n
=
n−1
¸
k=0

n
k

x
k
∆
1
x
=
1
x + 1
−
1
x
= −
1
x(x + 1)
∆

x
m

=

x + 1
m

−

x
m

=

x
m−1

The last example makes use of the recurrence relation
for binomial coefficients. An important observation
74 CHAPTER 6. FORMAL METHODS
concerns the behavior of the difference operator with
respect to the falling factorial:
∆x
m
= (x + 1)
m
−x
m
= (x + 1)x(x −1)
(x −m+ 2) −x(x −1) (x −m+ 1) =
= x(x −1) (x −m+ 2)(x + 1 −x +m−1) =
= mx
m−1
This is analogous to the usual rule for the differenti-
ation operator applied to x
m
:
Dx
m
= mx
m−1
As we shall see, many formal properties of the dif-
ference operator are similar to the properties of the
differentiation operator. The rˆole of the powers x
m
is
however taken by the falling factorials, which there-
fore assume a central position in the theory of finite
operators.
The following general rules are rather obvious:
∆(αf(x) +βg(x)) = α∆f(x) +β∆g(x)
∆(f(x)g(x)) =
= E(f(x)g(x)) −f(x)g(x) =
= f(x + 1)g(x + 1) −f(x)g(x) =
= f(x + 1)g(x + 1) −f(x)g(x + 1) +
+f(x)g(x + 1) −f(x)g(x) =
= ∆f(x)Eg(x) +f(x)∆g(x)
resembling the differentiation rule D(f(x)g(x)) =
(Df(x))g(x) +f(x)Dg(x). In a similar way we have:
∆
f(x)
g(x)
=
f(x + 1)
g(x + 1)
−
f(x)
g(x)
=
=
f(x + 1)g(x) −f(x)g(x)
g(x)g(x + 1)
+
+
f(x)g(x) −f(x)g(x + 1)
g(x)g(x + 1)
=
=
(∆f(x))g(x) −f(x)∆g(x)
g(x)Eg(x)
The difference operator can be iterated:
∆
2
f(x) = ∆∆f(x) = ∆(f(x + 1) −f(x)) =
= f(x + 2) −2f(x + 1) +f(x).
From a formal point of view, we have:
∆
2
= (E −1)
2
= E
2
−2E + 1
and in general:
∆
n
= (E −1)
n
=
n
¸
k=0

n
k

(−1)
n−k
E
k
=
= (−1)
n
n
¸
k=0

n
k

(−E)
k
This is a very important formula, and it is the first
example for the interest of combinatorics and gener-
ating functions in the theory of finite operators. In
fact, let us iterate ∆ on f(x) = 1/x:
∆
2
1
x
=
−1
(x + 1)(x + 2)
+
1
x(x + 1)
=
=
−x +x + 2
x(x + 1)(x + 2)
=
2
x(x + 1)(x + 2)
∆
n
1
x
=
(−1)
n
n!
x(x + 1) (x +n)
as we can easily show by mathematical induction. In
fact:
∆
n+1
1
x
=
(−1)
n
n!
(x + 1) (x +n + 1)
−
−
(−1)
n
n!
x(x + 1) (x +n)
=
=
(−1)
n+1
(n + 1)!
x(x + 1) (x +n + 1)
The formula for ∆
n
now gives the following identity:
∆
n
1
x
=
n
¸
k=0

n
k

(−1)
n−k
E
k
1
x
By multiplying everything by (−1)
n
this identity can
be written as:
n!
x(x + 1) (x +n)
=
1
x

x+n
n
=
n
¸
k=0

n
k

(−1)
k
x +k
and therefore we have both a way to express the in-
verse of a binomial coefficient as a sum and an expres-
sion for the partial fraction expansion of the polyno-
mial x(x + 1) (x +n) inverse.
6.8 Shift and Difference Oper-
ators - Example I
As the difference operator ∆ can be expressed in
terms of the shift operator E, so E can be expressed
in terms of ∆:
E = ∆ + 1
This rule can be iterated, giving the summation for-
mula:
E
n
= (∆ + 1)
n
=
n
¸
k=0

n
k

∆
k
which can be seen as the “dual” formula of the one
already considered:
∆
n
=
n
¸
k=0

n
k

(−1)
n−k
E
k
6.8. SHIFT AND DIFFERENCE OPERATORS - EXAMPLE I 75
The evaluation of the successive differences of any
function f(x) allows us to state and prove two identi-
ties, which may have combinatorial significance. Here
we record some typical examples; we mark with an
asterisk the cases when ∆
0
f(x) = If(x).
1) The function f(x) = 1/x has already been de-
veloped, at least partially:
∆
1
x
=
−1
x(x + 1)
∆
n
1
x
=
(−1)
n
n!
x(x + 1) (x +n)
(−1)
n
x

x +n
n

−1
¸
k

n
k

(−1)
k
x +k
=
n!
x(x + 1) (x +n)
=
=
1
x

x +n
n

−1
¸
k

n
k

(−1)
k

x +k
k

−1
=
x
x +n
.
2
∗
) A somewhat similar situation, but a bit more
complex:
∆
p +x
m+x
=
m−p
(m+x)(m+x + 1)
∆
n
p +x
m+x
=
m−p
m+x
(−1)
n−1

m+x +n
n

−1
¸
k

n
k

(−1)
k
p +k
m+k
=
m−p
m

m+n
n

−1
(n > 0)
¸
k

n
k

m+k
k

−1
(−1)
k
=
m
m+n
(see above).
3) Another version of the first example:
∆
1
px +m
=
−p
(px +m)(px +p +m)
∆
n
1
px +m
=
=
(−1)
n
n!p
n
(px +m)(px +p +m) (px +np +m)
According to this rule, we should have: ∆
0
(p +
x)/(m + x) = (p − m)(m + x); in the second next
sum, however, we have to set ∆
0
= I, and therefore
we also have to subtract 1 from both members in or-
der to obtain a true identity; a similar situation arises
whenever we have ∆
0
= I.
¸
k

n
k

(−1)
k
pk +m
=
n!p
n
m(m+p) (m+np)
¸
k

n
k

(−1)
k
k!p
k
m(m+p) (m+pk)
=
1
pn +m
4
∗
) A case involving the harmonic numbers:
∆H
x
=
1
x + 1
∆
n
H
x
=
(−1)
n−1
n

x +n
n

−1
¸
k

n
k

(−1)
k
H
x+k
= −
1
n

x +n
n

−1
(n > 0)
n
¸
k=1

n
k

(−1)
k−1
k

x +k
k

−1
= H
x+n
−H
x
where is to be noted the case x = 0.
5
∗
) A more complicated case with the harmonic
numbers:
∆xH
x
= H
x
+ 1
∆
n
xH
x
=
(−1)
n
n −1

x +n −1
n −1

−1
¸
k

n
k

(−1)
k
(x +k)H
x+k
=
=
1
n −1

x +n −1
n −1

−1
¸
k

n
k

(−1)
k
k −1

x +k −1
k −1

−1
=
= (x +n) (H
x+n
−H
x
) −n
6) Harmonic numbers and binomial coefficients:
∆

x
m

H
x
=

x
m−1

H
x
+
1
m

∆
n

x
m

H
x
=

x
m−n

(H
x
+H
m
−H
m−n
)
¸
k

n
k

(−1)
k

x +k
m

H
x+k
=
= (−1)
n

x
m−n

(H
x
+H
m
−H
m−n
)
¸
k

n
k

x
m−k

(H
x
+H
m
−H
m−k
) =
=

x +n
m

H
x+n
and by performing the sums on the left containing
H
x
and H
m
:
¸
k

n
k

x
m−k

H
m−k
=
=

x +n
m

(H
x
+H
m
−H
x+n
)
76 CHAPTER 6. FORMAL METHODS
7) The function ln(x) can be inserted in this group:
∆ln(x) = ln

x + 1
x

∆
n
ln(x) = (−1)
n
¸
k

n
k

(−1)
k
ln(x +k)
¸
k

n
k

(−1)
k
ln(x +k) =
=
¸
k

n
k

(−1)
k
ln(x +k)
n
¸
k=0
k
¸
j=0
(−1)
k+j

n
k

k
j

ln(x +j) = ln(x +n)
Note that the last but one relation is an identity.
6.9 Shift and Difference Oper-
ators - Example II
Here we propose other examples of combinatorial
sums obtained by iterating the shift and difference
operators.
1) The typical sum containing the binomial coeffi-
cients: Newton’s binomial theorem:
∆p
x
= (p −1)p
x
∆
n
p
x
= (p −1)
n
p
x
¸
k

n
k

(−1)
k
p
k
= (p −1)
n
¸
k

n
k

(p −1)
k
= p
n
2) Two sums involving Fibonacci numbers:
∆F
x
= F
x−1
∆
n
F
x
= F
x−n
¸
k

n
k

(−1)
k
F
x+k
= (−1)
n
F
x−n
¸
k

n
k

F
x−k
= F
x+n
3) Falling factorials are an introduction to binomial
coefficients:
∆x
m
= mx
m−1
∆
n
x
m
= m
n
x
m−n
¸
k

n
k

(−1)
k
(x +k)
m
= (−1)
n
m
n
x
m−n
¸
k

n
k

m
k
x
m−k
= (x +n)
m
4) Similar sums hold for raising factorials:
∆x
m
= m(x + 1)
m−1
∆
n
x
m
= m
n
(x +n)
m−n
¸
k

n
k

(−1)
k
(x +k)
m
= (−1)
n
m
n
(x +n)
m−n
¸
k

n
k

m
k
(x +k)
m−k
= (x +n)
m
5) Two sums involving the binomial coefficients:
∆

x
m

=

x
m−1

∆
n

x
m

=

x
m−n

¸
k

n
k

(−1)
k

x +k
m

= (−1)
n

x
m−n

¸
k

n
k

x
m−k

=

x +n
m

6) Another case with binomial coefficients:
∆

p +x
m+x

=

p +x
m+x + 1

∆
n

p +x
m+x

=

p +x
m+x +n

¸
k

n
k

(−1)
k

p +k
m+k

= (−1)
n

p
m+n

¸
k

n
k

p
m+k

=

p +n
m+n

7) And now a case with the inverse of a binomial
coefficient:
∆

x
m

−1
= −
m
m−1

x + 1
m+ 1

−1
∆
n

x
m

−1
= (−1)
n
m
m+n

x +n
m+n

−1
¸
k

n
k

(−1)
k

x +k
m

−1
=
m
m+n

x +n
m+n

−1
¸
k

n
k

(−1)
k
m
m+k

x +k
m+k

−1
=

x +n
m

−1
6.10. THE ADDITION OPERATOR 77
8) Two sums with the central binomial coefficients:
∆
1
4
x

2x
x

=
1
2(x + 1)
1
4
x

2x
x

∆
n
1
4
x

2x
x

=
(−1)
n
(2n)!
n!(x + 1) (x +n)
1
4
n

2n
n

¸
k

n
k

(−1)
k

2x + 2k
x +k

1
4
k
=
=
(2n)!
n!(x + 1) (x +n)
1
4
n

2x
x

=
=
1
4
n

2n
n

x +n
n

−1

2x
x

¸
k

n
k

(−1)
k
(2k)!
k!(x + 1) (x +k)
1
4
k

2x
x

=
=
1
4
n

2x + 2n
x +n

9) Two sums with the inverse of the central bino-
mial coefficients:
∆4
x

2x
x

−1
=
4
x
2x + 1

2x
x

−1
∆
n
4
x

2x
x

−1
=
=
1
2n −1
(−1)
n−1
(2n)!4
x
2
n
n!(2x + 1) (2x + 2n −1)

2x
x

−1
¸
k

n
k

(−1)
k
4
k

2x + 2k
x +k

−1
=
=
1
2n −1
(2n)!
2
n
n!(2x + 1) (2x + 2n −1)

2x
x

−1
¸
k

n
k

1
2k −1
(2k)!(−1)
k−1
2
k
k!(2x + 1) (2x + 2k −1)
=
= 4
n

2x + 2n
x +n

−1

2x
x

−1
6.10 The Addition Operator
The addition operator S is analogous to the difference
operator:
S = E + 1
and in fact a simple connection exists between the
two operators:
S(−1)
x
f(x) = (−1)
x+1
f(x + 1) + (−1)
x
f(x) =
= (−1)
x+1
(f(x + 1) −f(x)) =
= (−1)
x−1
∆f(x)
Because of this connection, the addition operator has
not widely considered in the literature, and the sym-
bol S only is used here for convenience. Likewise the
difference operator, the addition operator can be it-
erated and often produces interesting combinatorial
sums according to the rules:
S
n
= (E + 1)
n
=
¸
k

n
k

E
k
E
n
= (S −1)
n
=
¸
k

n
k

(−1)
n−k
S
k
Some examples are in order here:
1) Fibonacci numbers are typical:
SF
m
= F
m+1
+F
m
= F
m+2
S
n
F
m
= F
m+2n
¸
k

n
k

F
m+k
= F
m+2n
¸
k

n
k

(−1)
k
F
m+2k
= (−1)
n
F
m+n
2) Here are the binomial coefficients:
S

m
x

=

m+ 1
x + 1

S
n

m
x

=

m+n
x +n

¸
k

n
k

m
x +k

=

m+n
x +n

¸
k

n
k

(−1)
k

m+k
x +k

= (−1)
n

m
x +n

3) And finally the inverse of binomial coefficients:
S

m
x

−1
=
m+ 1
m

m−1
x

−1
S
n

m
x

=
m+ 1
m−n + 1

m−n
x

−1
¸
k

n
k

m
x +k

−1
=
m+ 1
m−n + 1

m+n
x

−1
¸
k

n
k

(−1)
k
m−k + 1

m+k
x

−1
=
1
m+ 1

m
x +n

−1
.
We can obviously invent as many expressions as we
desire and, correspondingly, may obtain some sum-
mation formulas of combinatorial interest. For ex-
ample:
S∆ = (E + 1)(E −1) = E
2
−1 =
= (E −1)(E + 1) = ∆S
78 CHAPTER 6. FORMAL METHODS
This derivation shows that the two operators S and
∆ commute. We can directly verify this property:
S∆f(x) = S(f(x + 1) −f(x)) =
= f(x + 2) −f(x) = (E
2
−1)f(x)
∆Sf(x) = ∆(f(x + 1) +f(x)) =
= f(x + 2) −f(x) = (E
2
−1)f(x)
Consequently, we have the two summation formulas:
∆
n
S
n
= (E
2
−1)
n
=
¸
k

n
k

(−1)
n−k
E
2k
E
2n
= (∆S + 1)
n
=
¸
k

n
k

∆
k
S
k
A simple example is offered by the Fibonacci num-
bers:
∆SF
m
= F
m+1
(∆S)
n
F
m
= ∆
n
S
n
F
m
= F
m+n
¸
k

n
k

(−1)
k
F
m+2k
= (−1)
n
F
m+n
¸
k

n
k

F
m+k
= F
m+2n
but these identities have already been proved using
the addition operator S.
6.11 Definite and Indefinite
summation
The following result is one of the most important rule
connecting the finite operator method and combina-
torial sums:
n
¸
k=0
E
k
= [z
n
]
1
1 −z
1
1 −Ez
=
= [z
n
]
1
(E −1)z

1
1 −Ez
−
1
1 −z

=
=
1
E −1
[z
n+1
]

1
1 −Ez
−
1
1 −z

=
=
E
n+1
−1
E −1
= (E
n+1
−1)∆
−1
We observe that the operator E commutes with the
indeterminate z, which is constant with respect to
the variable x, on which E operates. The rule above
is called the rule of definite summation; the operator
∆
−1
is called indefinite summation and is often de-
noted by Σ. In order to make this point clear, let us
consider any function f(x) and suppose that a func-
tion g(x) exists such that ∆g(x) = f(x). Hence we
have ∆
−1
f(x) = g(x) and the rule of definite sum-
mation immediately gives:
n
¸
k=0
f(x +k) = g(x +n + 1) −g(x)
This is analogous to the rule of definite integration.
In fact, the operator of indefinite integration

dx
is inverse of the differentiation operator D, and if
f(x) is any function, a primitive function for f(x)
is any function ˆ g(x) such that Dˆ g(x) = f(x) or
D
−1
f(x) =

f(x)dx = ˆ g(x). The fundamental the-
orem of the integral calculus relates definite and in-
definite integration:

b
a
f(x)dx = ˆ g(b) − ˆ g(a)
The formula for definite summation can be written
in a similar way, if we consider the integer variable k
and set a = x and b = x +n + 1:
b−1
¸
k=a
f(k) = g(b) −g(a)
These facts create an analogy between ∆
−1
and
D
−1
, or Σ and

dx, which can be stressed by con-
sidering the formal properties of Σ. First of all, we
observe that g(x) = Σf(x) is not uniquely deter-
mined. If C(x) is any function periodic of period
1, i.e., C(x +k) = C(x), ∀k ∈ Z, we have:
∆(g(x) +C(x)) = ∆g(x) + ∆C(x) = ∆g(x)
and therefore:
Σf(x) = g(x) +C(x)
When f(x) is a sequence and only is defined for inte-
ger values of x, the function C(x) reduces to a con-
stant, and plays the same rˆole as the integration con-
stant in the operation of indefinite integration.
The operator Σ is obviously linear:
Σ(αf(x) +βg(x)) = αΣf(x) +βΣg(x)
This is proved by applying the operator ∆ to both
sides.
An important property is summation by parts, cor-
responding to the well-known rule of the indefinite
integration operator. Let us begin with the rule for
the difference of a product, which we proved earlier:
∆(f(x)g(x)) = ∆f(x)Eg(x) +f(x)∆g(x)
By applying the operator Σ to both sides and ex-
changing terms:
Σ(f(x)∆g(x)) = f(x)g(x) −Σ(∆f(x)Eg(x))
6.12. DEFINITE SUMMATION 79
This formula allows us to change a sum of products,
of which we know that the second factor is a differ-
ence, into a sum involving the difference of the first
factor. The transformation can be convenient every
time when the difference of the first factor is simpler
than the difference of the second factor. For example,
let us perform the following indefinite summation:
Σx

x
m

= Σx∆

x
m+ 1

=
= x

x
m+ 1

−Σ(∆x)

x + 1
m+ 1

=
= x

x
m+ 1

−Σ

x + 1
m+ 1

=
= x

x
m+ 1

−

x + 1
m+ 2

Obviously, this indefinite sum can be transformed
into a definite sum by using the first result in this
section:
b
¸
k=a
k

k
m

= (b + 1)

b + 1
m+ 1

−
− a

a
m+ 1

−

b + 2
m+ 2

+

a + 1
m+ 2

and for a = 0 and b = n:
n
¸
k=0
k

k
m

= (n + 1)

n + 1
m+ 1

−

n + 2
m+ 2

6.12 Definite Summation
In a sense, the rule:
n
¸
k=0
E
k
= (E
n+1
−1)∆
−1
= (E
n+1
−1)Σ
is the most important result of the operator method.
In fact, it reduces the sum of the successive elements
in a sequence to the computation of the indefinite
sum, and this is just the operator inverse of the
difference. Unfortunately, ∆
−1
is not easy to com-
pute and, apart from a restricted number of cases,
there is no general rule allowing us to guess what
∆
−1
f(x) = Σf(x) might be. In this rather pes-
simistic sense, the rule is very fine, very general and
completely useless.
However, from a more positive point of view, we
can say that whenever we know, in some way or an-
other, an expression for ∆
−1
f(x) = Σf(x), we have
solved the problem of finding
¸
n
k=0
f(x + k). For
example, we can look at the differences computed in
the previous sections and, for each of them, obtain
the Σ of some function; in this way we immediately
have a number of sums. The negative point is that,
sometimes, we do not have a simple function and,
therefore, the sum may not have any combinatorial
interest.
Here is a number of identities obtained by our pre-
vious computations.
1) We have again the partial sums of the geometric
series:
∆
−1
p
x
=
p
x
p −1
= Σp
x
n
¸
k=0
p
x+k
=
p
x+n+1
−p
x
p −1
n
¸
k=0
p
k
=
p
n+1
−1
p −1
(x = 0)
2) The sum of consecutive Fibonacci numbers:
∆
−1
F
x
= F
x+1
= ΣF
x
n
¸
k=0
F
x+k
= F
x+n+2
−F
x+1
n
¸
k=0
F
k
= F
n+2
−1 (x = 0)
3) The sum of consecutive binomial coefficients with
constant denominator:
∆
−1

x
m

=

x
m+ 1

= Σ

x
m

n
¸
k=0

x +k
m

=

x +n + 1
m+ 1

−

x
m+ 1

n
¸
k=0

k
m

=

n + 1
m+ 1

(x = 0)
4) The sum of consecutive binomial coefficients:
∆
−1

p +x
m+x

=

p +x
m+x −1

= Σ

p +x
m+x

n
¸
k=0

p +k
m+k

=

p +n + 1
m+n

−

p
m−1

5) The sum of falling factorials:
∆
−1
x
m
=
1
m+ 1
x
m+1
= Σx
m
n
¸
k=0
(x +k)
m
=
(x +n + 1)
m+1
−x
m+1
m+ 1
n
¸
k=0
k
m
=
1
m+ 1
(n + 1)
m+1
(x = 0).
6) The sum of raising factorials:
∆
−1
x
m
=
1
m+ 1
(x −1)
m+1
= Σx
m
80 CHAPTER 6. FORMAL METHODS
n
¸
k=0
(x +k)
m
=
(x +n)
m+1
−(x −1)
m+1
m+ 1
n
¸
k=0
k
m
=
1
m+ 1
n
m+1
(x = 0).
7) The sum of inverse binomial coefficients:
∆
−1

x
m

−1
=
m
m−1

x −1
m−1

−1
= Σ

x
m

n
¸
k=0

x +k
m

−1
=
m
m−1


x −1
m−1

−1
−

x +n
m−1

−1

.
8) The sum of harmonic numbers. Since 1 = ∆x, we
have:
∆
−1
H
x
= xH
x
−x = ΣH
x
n
¸
k=0
H
x+k
= (x +n + 1)H
x+n+1
−xH
x
−(n + 1)
n
¸
k=0
H
k
= (n + 1)H
n+1
−(n + 1) =
= (n + 1)H
n
−n (x = 0).
6.13 The Euler-McLaurin Sum-
mation Formula
One of the most striking applications of the finite
operator method is the formal proof of the Euler-
McLaurin summation formula. The starting point is
the Taylor theorem for the series expansion of a func-
tion f(x) ∈ C
∞
, i.e., a function having derivatives of
any order. The usual form of the theorem:
f(x +h) = f(x) +
h
1!
f
′
(x) +
h
2
2!
f
′′
(x) +
+ +
h
n
n!
f
(n)
(x) +
can be interpreted in the sense of operators as a result
connecting the shift and the differentiation operators.
In fact, for h = 1, it can be written as:
Ef(x) = If(x) +
Df(x)
1!
+
D
2
f(x)
2!
+
and therefore as a relation between operators:
E = 1 +
D
1!
+
D
2
2!
+ +
D
n
n!
+ = e
D
This formal identity relates the finite operator E and
the infinitesimal operator D, and subtracting 1 from
both sides it can be formulated as:
∆ = e
D
−1
By inverting, we have a formula for the Σ operator:
Σ =
1
e
D
−1
=
1
D

D
e
D
−1

Now, we recognize the generating function of the
Bernoulli numbers, and therefore we have a devel-
opment for Σ:
Σ =
1
D

B
0
+
B
1
1!
D +
B
2
2!
D
2
+

=
= D
−1
−
1
2
I +
1
12
D −
1
720
D
3
+
+
1
30240
D
5
−
1
1209600
D
7
+ .
This is not a series development since, as we know,
the Bernoulli numbers diverge to infinity. We have a
case of asymptotic development, which only is defined
when we consider a limited number of terms, but in
general diverges if we let the number of terms go to
infinity. The number of terms for which the sum ap-
proaches its true value depends on the function f(x)
and on the argument x.
From the indefinite we can pass to the definite sum
by applying the general rule of Section 6.12. Since
D
−1
=

dx, we immediately have:
n−1
¸
k=0
f(k) =

n
0
f(x) dx −
1
2
[f(x)]
n
0
+
+
1
12
[f
′
(x)]
n
0
−
1
720
[f
′′′
(x)]
n
0
+
and this is the celebrated Euler-McLaurin summation
formula. It expresses a sum as a function of the in-
tegral and the successive derivatives of the function
f(x). In this sense, the formula can be seen as a
method for approximating a sum by means of an in-
tegral or, vice versa, for approximating an integral by
means of a sum, and this was just the point of view
of the mathematicians who first developed it.
As a simple but very important example, let us
find an asymptotic development for the harmonic
numbers H
n
. Since H
n
= H
n−1
+ 1/n, the Euler-
McLaurin formula applies to H
n−1
and to the func-
tion f(x) = 1/x, giving:
H
n−1
=

n
1
dx
x
−
1
2
¸
1
x

n
1
+
1
12
¸
−
1
x
2

n
1
−
−
1
720
¸
−
6
x
4

n
1
+
1
30240
¸
−
120
x
6

n
1
+
= ln n −
1
2n
+
1
2
−
1
12n
2
+
1
12
+
1
120n
4
−
−
1
120
−
1
256n
6
+
1
252
+
6.14. APPLICATIONS OF THE EULER-MCLAURIN FORMULA 81
In this expression a number of constants appears, and
they can be summed together to form a constant γ,
provided that the sum actually converges. However,
we observe that as n → ∞ this constant γ is the
Euler-Mascheroni constant:
lim
n→∞
(H
n−1
−ln n) = γ = 0.577215664902 . . .
By adding 1/n to both sides of the previous relation,
we eventually find:
H
n
= ln n +γ +
1
2n
−
1
12n
2
+
1
120n
4
−
1
252n
6
+
and this is the asymptotic expansion we were looking
for.
6.14 Applications of the Euler-
McLaurin Formula
As another application of the Euler-McLaurin sum-
mation formula, we now show the derivation of the
Stirling’s approximation for n!. The first step consists
in taking the logarithm of that quantity:
ln n! = ln 1 + ln 2 + ln 3 + + ln n
so that we are reduced to compute a sum and hence
to apply the Euler-McLaurin formula:
ln(n −1)! =
n−1
¸
k=1
ln k =

n
1
ln xdx −
1
2
[ln x]
n
1
+
+
1
12
¸
1
x

n
1
−
1
720
¸
2
x
3

n
1
+ =
= nln n −n + 1 −
1
2
ln n +
1
12n
−
−
1
12
−
1
360n
2
+
1
360
+ .
Here we have used the fact that

ln xdx = xln x −
x. At this point we can add lnn to both sides and
introduce a constant σ = 1 −1/12 +1/360 − It is
not by all means easy to determine directly the value
of σ, but by other approaches to the same problem
it is known that σ = ln
√
2π. Numerically, we can
observe that:
1−
1
12
+
1
360
= 0.919(4) and ln
√
2π ≈ 0.9189388.
We can now go on with our sum:
ln n! = nln n−n+
1
2
ln n+ln
√
2π+
1
12n
−
1
360n
3
+
To obtain the value of n! we only have to take expo-
nentials:
n! =
n
n
e
n
√
n
√
2π exp

1
12n

exp

−
1
360n
3


=
√
2πn
n
n
e
n

1 +
1
12n
+
1
288n
2
+




1 −
1
360n
3
+

=
=
√
2πn
n
n
e
n

1 +
1
12n
+
1
288n
2
−

.
This is the well-known Stirling’s approximation for
n!. By means of this approximation, we can also find
the approximation for another important quantity:

2n
n

=
(2n)!
n!
2
=
√
4πn

2n
e

2n
2πn

n
e

2n


1 +
1
24n
+
1
1142n
2
−

1 +
1
12n
+
1
288n
2
−

2
=
=
4
n
√
πn

1 −
1
8n
+
1
128n
2
+

.
Another application of the Euler-McLaurin sum-
mation formula is given by the sum
¸
n
k=1
k
p
, when
p is any integer constant different from −1, which is
the case of the harmonic numbers:
n−1
¸
k=0
k
p
=

n
0
x
p
dx −
1
2
[x
p
]
n
0
+
1
12

px
p−1

n
0
−
−
1
720

p(p −1)(p −2)x
p−3

n
0
+ =
=
n
p+1
p + 1
−
n
p
2
+
pn
p−1
12
−
−
p(p −1)(p −2)n
p−3
720
+ .
In this case the evaluation at 0 does not introduce
any constant. By adding n
p
to both sides, we have
the following formula, which only contains a finite
number of terms:
n
¸
k=0
k
p
=
n
p+1
p + 1
+
n
p
2
+
pn
p−1
12
−
−
p(p −1)(p −2)n
p−3
720
+
If p is not an integer, after ⌈p⌉ differentiations, we
obtain x
q
, where q < 0, and therefore we cannot
consider the limit 0. We proceed with the Euler-
McLaurin formula in the following way:
n−1
¸
k=1
k
p
=

n
1
x
p
dx −
1
2
[x
p
]
n
1
+
1
12

px
p−1

n
1
−
−
1
720

p(p −1)(p −2)x
p−3

n
1
+ =
=
n
p+1
p + 1
−
1
p + 1
−
1
2
n
p
+
1
2
+
pn
p−1
12
−
−
p
12
−
p(p −1)(p −2)n
p−3
720
+
82 CHAPTER 6. FORMAL METHODS
+
p(p −1)(p −2)
720

n
¸
k=1
k
p
=
n
p+1
p + 1
+
n
p
2
+
pn
p−1
12
−
−
p(p −1)(p −2)n
p−3
720
+ +K
p
.
The constant:
K
p
= −
1
p + 1
+
1
2
−
p
12
+
p(p −1)(p −2)
720
+
has a fundamental rˆole when the leading term
n
p+1
/(p + 1) does not increase with n, i.e., when
p < −1. In that case, in fact, the sum converges
to K
p
. When p > −1 the constant is less important.
For example, we have:
n
¸
k=1
√
k =
2
3
n
√
n +
√
n
2
+K
1/2
+
1
24
√
n
+
K
1/2
≈ −0.2078862 . . .
n
¸
k=1
1
√
k
= 2
√
n +K
−1/2
+
1
2
√
n
−
1
24n
√
n
+
K
−1/2
≈ −1.4603545 . . .
For p = −2 we find:
n
¸
k=1
1
k
2
= K
−2
−
1
n
+
1
2n
2
−
1
6n
3
+
1
30n
5
−
It is possible to show that K
−2
= π
2
/6 and therefore
we have a way to approximate the sum (see Section
2.7).
Chapter 7
Asymptotics
7.1 The convergence of power
series
In many occasions, we have pointed out that our ap-
proach to power series was purely formal. Because
of that, we always spoke of “formal power series”,
and never considered convergence problems. As we
have seen, a lot of things can be said about formal
power series, but now the moment has arrived that
we must turn to talk about the convergence of power
series. We will see that this allows us to evaluate the
asymptotic behavior of the coefficients f
n
of a power
series
¸
n
f
n
t
n
, thus solving many problems in which
the exact value of f
n
cannot be found. In fact, many
times, the asymptotic evaluation of f
n
can be made
more precise and an actual approximation of f
n
can
be found.
The natural setting for talking about convergence
is the field C of the complex numbers and therefore,
from now on, we will think of the indeterminate t
as of a variable taking its values from C. Obviously,
a power series f(t) =
¸
n
f
n
t
n
converges for some
t
0
∈ C iff f(t
0
) =
¸
n
f
n
t
n
0
< ∞, and diverges iff
lim
t→t0
f(t) = ∞. There are cases for which a series
neither converges nor diverges; for example, when t =
1, the series
¸
n
(−1)
n
t
n
does not tend to any limit,
finite or infinite. Therefore, when we say that a series
does not converge (to a finite value) for a given value
t
0
∈ C, we mean that the series in t
0
diverges or does
not tend to any limit.
A basic result on convergence is given by the fol-
lowing:
Theorem 7.1.1 Let f(t) =
¸
n
f
n
t
n
be a power se-
ries such that f(t
0
) converges for the value t
0
∈ C.
Then f(t) converges for every t
1
∈ C such that
[t
1
[ < [t
0
[.
Proof: If f(t
0
) < ∞ then an index N ∈ N ex-
ists such that for every n > N we have [f
n
t
n
0
[ ≤
[f
n
[[t
0
[
n
< M, for some finite M ∈ R. This means
[f
n
[ < M/[t
0
[
n
and therefore:





∞
¸
n=N
f
n
t
n
1





≤
∞
¸
n=N
[f
n
[[t
1
[
n
≤
≤
∞
¸
n=N
M
[t
0
[
n
[t
1
[
n
= M
∞
¸
n=N

[t
1
[
[t
0
[

n
< ∞
because the last sum is a geometric series with
[t
1
[/[t
0
[ < 1 by the hypothesis [t
1
[ < [t
0
[. Since the
first N terms obviously amount to a finite quantity,
the theorem follows.
In a similar way, we can prove that if the series
diverges for some value t
0
∈ C, then it diverges for
every value t
1
such that [t
1
[ > [t
0
[. Obviously, a
series can converge for the single value t
0
= 0, as it
happens for
¸
n
n!t
n
, or can converge for every value
t ∈ C, as for
¸
n
t
n
/n! = e
t
. In all the other cases,
the previous theorem implies:
Theorem 7.1.2 Let f(t) =
¸
n
f
n
t
n
be a power se-
ries; then there exists a non-negative number R ∈ R
or R = ∞ such that:
1. for every complex number t
0
such that [t
0
[ < R
the series (absolutely) converges and, in fact, the
convergence is uniform in every circle of radius
ρ < R;
2. for every complex number t
0
such that [t
0
[ > R
the series does not converge.
The uniform convergence derives from the previous
proof: the constant M can be made unique by choos-
ing the largest value for all the t
0
such that [t
0
[ ≤ ρ.
The value of R is uniquely determined and is called
the radius of convergence for the series. From the
proof of the theorem, for r < R we have [f
n
[r
n
≤
M or
n

[f
n
[ ≤
n
√
M/r → 1/r; this implies that
limsup
n

[f
n
[ ≤ 1/R. Besides, for r > R we
have [f
n
[r
n
≥ 1 for infinitely many n; this implies
limsup
n

[f
n
[ ≥ 1/R, and therefore we have the fol-
lowing formula for the radius of convergence:
1
R
= lim sup
n→∞
n

[f
n
[.
83
84 CHAPTER 7. ASYMPTOTICS
This result is the basis for our considerations on the
asymptotics of a power series coefficients. In fact, it
implies that, as a first approximation, [f
n
[ grows as
1/R
n
. However, this is a rough estimate, because it
can also grow as n/R
n
or 1/(nR
n
), and many possi-
bilities arise, which can make more precise the basic
approximation; the next sections will be dedicated to
this problem. We conclude by noticing that if:
lim
n→∞




f
n+1
f
n




= S
then R = 1/S is the radius of convergence of the
series.
7.2 The method of Darboux
Newton’s rule is the basis for many considerations on
asymptotics. In practice, we used it to prove that
F
n
∼ φ
n
/
√
5, and many other proofs can be per-
formed by using Newton’s rule together with the fol-
lowing theorem, whose relevance was noted by Ben-
der and, therefore, will be called Bender’s theorem:
Theorem 7.2.1 Let f(t) = g(t)h(t) where f(t),
g(t), h(t) are power series and h(t) has a ra-
dius of convergence larger than f(t)’s (which there-
fore equals the radius of convergence of g(t)); if
lim
n→∞
g
n
/g
n+1
= b and h(b) = 0, then:
f
n
∼ h(b)g
n
Let us remember that if g(t) has positive real coef-
ficients, then g
n
/g
n+1
tends to the radius of conver-
gence of g(t). The proof of this theorem is omitted
here; instead, we give a simple example. Let us sup-
pose we wish to find the asymptotic value for the
Motzkin numbers, whose generating function is:
µ(t) =
1 −t −
√
1 −2t −3t
2
2t
2
.
For n ≥ 2 we obviously have:
µ
n
= [t
n
]
1 −t −
√
1 −2t −3t
2
2t
2
=
= −
1
2
[t
n+2
]
√
1 +t (1 −3t)
1/2
.
We now observe that the radius of convergence of
µ(t) is R = 1/3, which is the same as the radius of
g(t) = (1−3t)
1/2
, while h(t) =
√
1 +t has 1 as radius
of convergence; therefore we have µ
n
/µ
n+1
→ 1/3 as
n → ∞. By Bender’s theorem we find:
µ
n
∼ −
1
2

4
3
[t
n+2
](1 −3t)
1/2
=
= −
√
3
3

1/2
n + 2

(−3)
n+2
=
=
√
3
3(2n + 3)

2n + 4
n + 2

3
4

n+2
.
This is a particular case of a more general re-
sult due to Darboux and known as Darboux’ method.
First of all, let us show how it is possible to obtain an
approximation for the binomial coefficient

γ
n

, when
γ ∈ C is a fixed number and n is large. We begin
by proving the following formula for the ratio of two
large values of the Γ function (a, b are two small pa-
rameters with respect to n):
Γ(n +a)
Γ(n +b)
=
= n
a−b

1 +
(a −b)(a +b −1)
2n
+O

1
n
2

.
Let us apply the Stirling formula for the Γ function:
Γ(n +a)
Γ(n +b)
≈
≈

2π
n +a

n +a
e

n+a

1 +
1
12(n +a)




n +b
2π

e
n +b

n+b

1 −
1
12(n +b)

.
If we limit ourselves to the term in 1/n, the two cor-
rections cancellate each other and therefore we find:
Γ(n +a)
Γ(n +b)
≈

n +b
n +a
e
b−a
(n +a)
n+a
(n +b)
n+b
=
=

n +b
n +a
e
b−a
n
a−b
(1 +a/n)
n+a
(1 +b/n)
n+b
.
We now obtain asymptotic approximations in the fol-
lowing way:

n +b
n +a
=

1 +b/n

1 +a/n
≈
≈

1 +
b
2n


1 −
a
2n

≈ 1 +
b −a
2n
.

1 +
x
n

n+x
= exp

(n +x) ln

1 +
x
n

=
= exp

(n +x)

x
n
−
x
2
2n
2
+

=
= exp

x +
x
2
n
−
x
2
2n
+

=
= e
x

1 +
x
2
2n
+

.
Therefore, for our expression we have:
Γ(n +a)
Γ(n +b)
≈
≈
n
a−b
e
a−b
e
a
e
b

1 +
a
2
2n

1 −
b
2
2n

1 +
b −a
2n

=
= n
a−b

1 +
a
2
−b
2
−a +b
2n
+O

1
n
2

.
We are now in a position to prove the following:
7.3. SINGULARITIES: POLES 85
Theorem 7.2.2 Let f(t) = h(t)(1 − αt)
γ
, for some
γ which is not a positive integer, and h(t) having a
radius of convergence larger than 1/α. Then we have:
f
n
= [t
n
]f(t) ∼ h

1
α

γ
n

(−α)
n
=
α
n
h(1/α)
Γ(−γ)n
1+γ
.
Proof: We simply apply Bender’s theorem and the
formula for approximating the binomial coefficient:

γ
n

=
γ(γ −1) (γ −n + 1)
n!
=
=
(−1)
n
(n −γ −1)(n −γ −2) (1 −γ)(−γ)
Γ(n + 1)
.
By repeated applications of the recurrence formula
for the Γ function Γ(x + 1) = xΓ(x), we find:
Γ(n −γ) =
= (n −γ −1)(n −γ −2) (1 −γ)(−γ)Γ(−γ)
and therefore:

γ
n

=
(−1)
n
Γ(n −γ)
Γ(n + 1)Γ(−γ)
=
=
(−1)
n
Γ(−γ)
n
−1−γ

1 +
γ(γ + 1)
2n

from which the desired formula follows.
7.3 Singularities: poles
The considerations in the previous sections show how
important is to determine the radius of convergence
of a series, when we wish to have an approximation
of its coefficients. Therefore, we are now going to
look more closely to methods for finding the radius
of convergence of a given function f(t). In order to do
this, we have to distinguish between the function f(t)
and its series development, which will be denoted by
´
f(t). So, for example, we have f(t) = 1/(1 − t) and
´
f(t) = 1 +t +t
2
+t
3
+ . The series
´
f(t) represents
f(t) inside the circle of convergence, in the sense that
´
f(t) = f(t) for every t internal to this circle, but
´
f(t) can be different from f(t) on the border of the
circle or outside it, where actually the series does not
converge. Therefore, the radius of convergence can
be determined if we are able to find out values t
0
for
which f(t
0
) =
´
f(t
0
).
A first case is when t
0
is an isolated point for which
lim
t→t0
f(t) = ∞. In fact, in this case, for every t
such that [t[ > [t
0
[ the series
´
f(t) should diverge as
we have seen in the previous section, and therefore
´
f(t) must be different from f(t). This is the case of
f(t) = 1/(1−t), which goes to ∞ when t → 1. When
[t[ > 1, f(t) assumes a well-defined value while
´
f(t)
diverges.
We will call a singularity for f(t) every point t
0
∈ C
such that in every neighborhood of t
0
there is a t for
which f(t) and
´
f(t) behaves differently and a t
′
for
which f(t
′
) =
´
f(t
′
). Therefore, t
0
= 1 is a singularity
for f(t) = 1/(1 −t). Because our previous considera-
tions, the singularities of f(t) determine its radius of
convergence; on the other hand, no singularity can be
contained in the circle of convergence, and therefore
the radius of convergence is determined by the singu-
larity or singularities of smallest modulus. These will
be called dominating singularities and we observe ex-
plicitly that a function can have more than one dom-
inating singularity. For example, f(t) = 1/(1 − t
2
)
has t = 1 and t = −1 as dominating singularities, be-
cause [1[ = [ −1[. The radius of convergence is always
a non-negative real number and we have R = [t
0
[, if
t
0
is any one of the dominating singularities for f(t).
An isolated point t
0
for which f(t
0
) = ∞ is there-
fore a singularity for f(t); as we shall see, not every
singularity of f(t) is such that f(t) = ∞, but, for
the moment, let us limit ourselves to this case. The
following situation is very important: if f(t
0
) = ∞
and we set α = 1/t
0
, we will say that t
0
is a pole for
f(t) iff there exists a positive integer m such that:
lim
t→t0
(1 −αt)
m
f(t) = K < ∞ and K = 0.
The integer m is called the order of the pole. By
this definition, the function f(t) = 1/(1 − t) has a
pole of order 1 in t
0
= 1, while 1/(1 − t)
2
has a
pole of order 2 in t
0
= 1 and 1/(1 − 2t)
5
has a pole
of order 5 in t
0
= 1/2. A more interesting case is
f(t) = (e
t
− e)/(1 − t)
2
, which, notwithstanding the
(1 −t)
2
, has a pole of order 1 in t
0
= 1; in fact:
lim
t→1
(1 −t)
e
t
−e
(1 −t)
2
= lim
t→1
e
t
−e
1 −t
= lim
t→1
e
t
−1
= −e.
The generating function of Bernoulli numbers
f(t) = t/(e
t
− 1) has infinitely many poles. Observe
first that t = 0 is not a pole because:
lim
t→0
t
e
t
−1
= lim
t→0
1
e
t
= 1.
The denominator becomes 0 when e
t
= 1, and this
happens when t = 2kπi; in fact, e
2kπi
= cos 2kπ +
i sin 2kπ = 1. In that case, the dominating sin-
gularities are t
0
= 2πi and t
1
= −2πi. Finally,
the generating function of the ordered Bell numbers
f(t) = 1/(2−e
t
) has again an infinite number of poles
t = ln 2+2kπi; in this case the dominating singularity
is t
0
= ln 2.
We conclude this section by observing that if
f(t
0
) = ∞, not necessarily t
0
is a pole for f(t). In
86 CHAPTER 7. ASYMPTOTICS
fact, let us consider the generating function for the
central binomial coefficients f(t) = 1/
√
1 −4t. For
t
0
= 1/4 we have f(1/4) = ∞, but t
0
is not a pole of
order 1 because:
lim
t→1/4
1 −4t
√
1 −4t
= lim
t→1/4
√
1 −4t = 0
and the same happens if we try with (1 − 4t)
m
for
m > 1. As we shall see, this kind of singularity is
called “algebraic”. Finally, let us consider the func-
tion f(t) = exp(1/(1−t)), which goes to ∞ as t → 1.
In this case we have:
lim
t→1
(1 −t)
m
exp

1
1 −t

=
= lim
t→1
(1 −t)
m

1 +
1
1 −t
+
1
2(1 −t)
2
+

= ∞.
In fact, the first m − 1 terms tend to 0, the mth
term tends to 1/m!, but all the other terms go to ∞.
Therefore, t
0
= 1 is not a pole of any order. When-
ever we have a function f(t) for which a point t
0
∈ C
exists such that ∀m > 0: lim
t→t0
(1−t/t
0
)
m
f(t) = ∞,
we say that t
0
is an essential singularity for f(t). Es-
sential singularities are points at which f(t) goes to
∞ too fast; these singularities cannot be treated by
Darboux’ method and their study will be delayed un-
til we study the Hayman’s method.
7.4 Poles and asymptotics
Darboux’ method can be easily used to deal with
functions, whose dominating singularities are poles.
Actually, a direct application of Bender’s theorem is
sufficient, and this is the way we will use in the fol-
lowing examples.
Fibonacci numbers are easily approximated:
[t
n
]
t
1 −t −t
2
= [t
n
]
t
1 −
´
φt
1
1 −φt
∼
∼
¸
t
1 −
´
φt


t =
1
φ

[t
n
]
1
1 −φt
=
1
√
5
φ
n
.
Our second example concerns a particular kind of
permutations, called derangements (see Section 2.2).
A derangement is a permutation without any fixed
point. For n = 0 the empty permutation is consid-
ered a derangement, since no fixed point exists. For
n = 1, there is no derangement, but for n = 2 the
permutation (1 2), written in cycle notation, is ac-
tually a derangement. For n = 3 we have the two
derangements (1 2 3) and (1 3 2), and for n = 4 we
have a total of 9 derangements.
Let D
n
the number of derangements in P
n
; we can
count them in the following way: we begin by sub-
tracting from n!, the total number of permutations,
the number of permutations having at least a fixed
point: if the fixed point is 1, we have (n−1)! possible
permutations; if the fixed point is 2, we have again
(n − 1)! permutations of the other elements. There-
fore, we have a total of n(n − 1)! cases, giving the
approximation:
D
n
= n! −n(n −1)!.
This quantity is clearly 0 and this happens because
we have subtracted twice every permutation with at
least 2 fixed points: in fact, we subtracted it when
we considered the first and the second fixed point.
Therefore, we have now to add permutations with at
least two fixed points. These are obtained by choos-
ing the two fixed points in all the

n
2

possible ways
and then permuting the n − 2 remaining elements.
Thus we have the new approximation:
D
n
= n! −n(n −1)! +

n
2

(n −2)!.
In this way, however, we added twice permutations
with at least three fixed points, which have to be
subtracted again. We thus obtain:
D
n
= n! −n(n −1)! +

n
2

(n −2)! −

n
3

(n −3)!.
We can now go on with the same method, which is
called the inclusion exclusion principle, and eventu-
ally arrive to the final value:
D
n
= n! −n(n −1)! +

n
2

(n −2)! −
−

n
3

(n −3)! + =
=
n!
0!
−
n!
1!
+
n!
2!
−
n!
3!
+ = n!
n
¸
k=0
(−1)
k
k!
.
This formula checks with the previously found val-
ues. We obtain the exponential generating function
((D
n
/n!) by observing that the generic element in
the sum is the coefficient [t
n
]e
−t
, and therefore by
the theorem on the generating function for the par-
tial sums we have:
(

D
n
n!

=
e
−t
1 −t
.
In order to find the asymptotic value for D
n
, we
observe that the radius of convergence of 1/(1 − t)
is 1, while e
−t
converges for every value of t. By
Bender’s theorem we have:
D
n
n!
∼ e
−1
or D
n
∼
n!
e
.
7.5. ALGEBRAIC AND LOGARITHMIC SINGULARITIES 87
This value is indeed a very good approximation for
D
n
, which can actually be computed as the integer
nearest to n!/e.
Let us now see how Bender’s theorem is applied
to the exponential generating function of the ordered
Bell numbers. We have shown that the dominating
singularity is a pole at t = ln 2 which has order 1:
lim
t→ln 2
1 −t/ ln 2
2 −e
t
= lim
t→ln 2
−1/ ln 2
−e
t
=
1
2 ln 2
.
At this point we have:
[t
n
]
1
2 −e
t
= [t
n
]
1
1 −t/ ln 2
1 −t/ ln 2
2 −e
t
∼
∼
¸
1 −t/ ln 2
2 −e
t


t = ln 2

[t
n
]
1
1 −t/ ln 2
=
=
1
2
1
(ln 2)
n+1
and we conclude with the very good approximation
O
n
∼ n!/(2(ln 2)
n+1
).
Finally, we find the asymptotic approximation for
the Bernoulli numbers. The following statement is
very important when we have functions with several
dominating singularity:
Principle: If t
1
, t
2
, . . . , t
k
are all the dominating
singularities of a function f(t), then [t
n
]f(t) can be
found by summing all the contributions obtained by
independently considering the k singularities.
We already observed that ±2πi are the two domi-
nating singularities for the generating function of the
Bernoulli numbers; they are both poles of order 1:
lim
t→2πi
t(1 −t/2πi)
e
t
−1
= lim
t→2πi
1 −t/πi
e
t
= −1.
lim
t→−2πi
t(1 +t/2πi)
e
t
−1
= lim
t→−2πi
1 +t/πi
e
t
= −1.
Therefore we have:
[t
n
]
t
e
t
−1
= [t
n
]
1
1 −t/2πi
t(1 −t/2πi)
e
t
−1
∼ −
1
(2πi)
n
.
A similar result is obtained for the other pole; thus
we have:
B
n
n!
∼ −
1
(2πi)
n
−
1
(−2πi)
n
.
When n is odd, these two values are opposite in sign
and the result is 0; this confirms that the Bernoulli
numbers of odd index are 0, except for n = 1. When
n is even, say n = 2k, we have (2πi)
2k
= (−2πi)
2k
=
(−1)
k
(2π)
2k
; therefore:
B
2k
∼ −
2(−1)
k
(2k)!
(2π)
2k
.
This formula is a good approximation, also for small
values of n, and shows that Bernoulli numbers be-
come, in modulus, larger and larger as n increases.
7.5 Algebraic and logarithmic
singularities
Let us consider the generating function for the Cata-
lan numbers f(t) = (1 −
√
1 −4t)/(2t) and the cor-
responding power series
´
f(t) = 1 + t + 2t
2
+ 5t
3
+
14t
4
+ . Our choice of the − sign was motivated
by the initial condition of the recurrence C
n+1
=
¸
n
k=0
C
k
C
n−k
defining the Catalan numbers. This
is due to the fact that, when the argument is a pos-
itive real number, we can choose the positive value
as the result of a square root. In other words, we
consider the arithmetic square root instead of the al-
gebraic square root. This allows us to identify the
power series
´
f(t) with the function f(t), but when we
pass to complex numbers this is no longer possible.
Actually, in the complex field, a function containing
a square root is a two-valued function, and there are
two branches defined by the same expression. Only
one of these two branches coincides with the func-
tion defined by the power series, which is obviously a
one-valued function.
The points at which a square root becomes 0 are
special points; in them the function is one-valued, but
in every neighborhood the function is two-valued. For
the smallest in modulus among these points, say t
0
,
we must have the following situation: for t such that
[t[ < [t
0
[,
´
f(t) should coincide with a branch of f(t),
while for t such that [t[ > [t
0
[,
´
f(t) cannot converge.
In fact, consider a t ∈ R, t > [t
0
[; the expression un-
der the square root should be a negative real number
and therefore f(t) ∈ C`R; but
´
f(t) can only be a real
number or f(t) does not converge. Because we know
that when
´
f(t) converges we must have
´
f(t) = f(t),
we conclude that
´
f(t) cannot converge. This shows
that t
0
is a singularity for f(t).
Every kth root originates the same problem and
the function is actually a k-valued function; all the
values for which the argument of the root is 0 is a
singularity, called an algebraic singularity. They can
be treated by Darboux’ method or, directly, by means
of Bender’s theorem, which relies on Newton’s rule.
Actually, we already used this method to find the
asymptotic evaluation for the Motzkin numbers.
The same considerations hold when a function con-
tains a logarithm. In fact, a logarithm is an infinite-
valued function, because it is the inverse of the expo-
nential, which, in the complex field C, is a periodic
function:
e
t+2kπi
= e
t
e
2kπi
= e
t
(cos 2kπ +i sin 2kπ) = e
t
.
The period of e
t
is therefore 2πi and lnt is actually
ln t + 2kπi, for k ∈ Z. A point t
0
for which the ar-
gument of a logarithm is 0 is a singularity for the
88 CHAPTER 7. ASYMPTOTICS
corresponding function. In every neighborhood of t
0
,
the function has an infinite number of branches; this
is the only fact distinguishing a logarithmic singular-
ity from an algebraic one.
Let us suppose we have the sum:
S
n
= 1 +
2
2
+
4
3
+
8
4
+ +
2
n−1
n
=
1
2
n
¸
k=1
2
k
k
and we wish to compute an approximate value. The
generating function is:
(

1
2
n
¸
k=1
2
k
k

=
1
2
1
1 −t
ln
1
1 −2t
.
There are two singularities: t = 1 is a pole, while t =
1/2 is a logarithmic singularity. Since the latter has
smaller modulus, it is dominating and R = 1/2 is the
radius of convergence of the function. By Bender’s
theorem we have:
S
n
=
1
2
[t
n
]
1
1 −t
ln
1
1 −2t
∼
∼
1
2
1
1 −1/2
[t
n
] ln
1
1 −2t
=
2
n
n
.
This is not a very good approximation. In the next
section we will see how it can be improved.
7.6 Subtracted singularities
The methods presented in the preceding sections only
give the expression describing the general behavior of
the coefficients f
n
in the expansion
´
f(t) =
¸
∞
k=0
f
k
t
k
,
i.e., what is called the principal value for f
n
. Some-
times, this behavior is only achieved for very large
values of n, but for smaller values it is just a rough
approximation of the true value. Because of that,
we speak of “asymptotic evaluation” or “asymptotic
approximation”. When we need a true approxima-
tion, we should introduce some corrections, which
slightly modify the general behavior and more ac-
curately evaluate the true value of f
n
.
Many times, the following observation solves the
problem. Suppose we have found, by one of the
previous methods, that a function f(t) is such that
f(t) ∼ A(1−αt)
γ
, for some A, γ ∈ R, or, more in gen-
eral, f(t) ∼ g(t), for some function g(t) of which we
exactly know the coefficients g
n
. For example, this is
the case of ln(1/(1−αt)). Because f
n
∼ g
n
, the func-
tion h(t) = f(t) − g(t) has coefficients h
n
that grow
more slowly than f
n
; formally, since O(f
n
) = O(g
n
),
we must have h
n
= o(f
n
). Therefore, the quantity
g
n
+h
n
is a better approximation to f
n
than g
n
.
When f(t) has a pole t
0
of order m, the successive
application of this method of subtracted singularities
completely eliminates the singularity. Therefore, if
a second singularity t
1
exists such that [t
1
[ > [t
0
[,
we can express the coefficients f
n
in terms of t
1
as
well. When f(t) has a dominating algebraic singular-
ity, this cannot be eliminated, but the method of sub-
tracted singularities allows us to obtain corrections to
the principal value. Formally, by the successive ap-
plication of this method, we arrive to the following
results. If t
0
is a dominating pole of order m and
α = 1/t
0
, then we find the expansion:
f(t) =
A
−m
(1 −αt)
m
+
A
−m+1
(1 −αt)
m−1
+
A
−m+2
(1 −αt)
m−2
+ .
If t
0
is a dominating algebraic singularity and f(t) =
h(t)(1 − αt)
p/m
, where α = 1/t
0
and h(t) has a ra-
dius of convergence larger than [t
0
[, then we find the
expansion:
f(t) = A
p
(1 −αt)
p/m
+A
p−1
(1 −αt)
(p−1)/m
+
+A
p−2
(1 −αt)
(p−2)/m
+ .
Newton’s rule can obviously be used to pass from
these expansions to the asymptotic value of f
n
.
The same method of subtracted singularities can be
used for a logarithmic singularity. Let us consider as
an example the sum S
n
=
¸
n
k=1
2
k−1
/k, introduced
in the previous section. We found the principal value
S
n
∼ 2
n
/n by studying the generating function:
1
2
1
1 −t
ln
1
1 −2t
∼ ln
1
1 −2t
.
Let us therefore consider the new function:
h(t) =
1
2
1
1 −t
ln
1
1 −2t
−ln
1
1 −2t
=
= −
1 −2t
2(1 −t)
ln
1
1 −2t
.
The generic term h
n
should be significantly less than
f
n
; the factor (1 − 2t) actually reduces the order of
growth of the logarithm:
−[t
n
]
1 −2t
2(1 −t)
ln
1
1 −2t
=
= −
1
2(1 −1/2)
[t
n
](1 −2t) ln
1
1 −2t
=
= −

2
n
n
−2
2
n−1
n −1

=
2
n
n(n −1)
.
Therefore, a better approximation for S
n
is:
S
n
=
2
n
n
+
2
n
n(n −1)
=
2
n
n −1
.
The reader can easily verify that this correction
greatly reduces the error in the evaluation of S
n
.
7.8. HAYMAN’S METHOD 89
A further correction can now be obtained by con-
sidering:
k(t) =
1
2
1
1 −t
ln
1
1 −2t
−
−ln
1
1 −2t
+ (1 −2t) ln
1
1 −2t
=
=
(1 −2t)
2
2(1 −t)
ln
1
1 −2t
which gives:
k
n
=
1
2(1 −1/2)
[t
n
](1 −2t)
2
ln
1
1 −2t
=
=
2
n
n
−4
2
n−1
n −1
+ 4
2
n−2
n −2
=
2
n+1
n(n −1)(n −2)
.
This correction is still smaller, and we can write:
S
n
∼
2
n
n −1

1 +
2
n(n −2)

.
In general, we can obtain the same results if we ex-
pand the function h(t) in f(t) = g(t)h(t), h(t) with a
radius of convergence larger than that of f(t), around
the dominating singularity. This is done in the fol-
lowing way:
1
2(1 −t)
=
1
1 + (1 −2t)
=
= 1 −(1 −2t) + (1 −2t)
2
−(1 −2t)
3
+ .
This implies:
1
2(1 −t)
ln
1
1 −2t
= ln
1
1 −2t
−
−(1 −2t) ln
1
1 −2t
+ (1 −2t)
2
ln
1
1 −2t
−
and the result is the same as the one previously ob-
tained by the method of subtracted singularities.
7.7 The asymptotic behavior of
a trinomial square root
In many problems we arrive to a generating function
of the form:
f(t) =
p(t) −

(1 −αt)(1 −βt)
rt
m
or:
g(t) =
q(t)

(1 −αt)(1 −βt)
.
In the former case, p(t) is a correcting polynomial,
which has no effect on f
n
, for n sufficiently large,
and therefore we have:
f
n
= −
1
r
[t
n+m
]

(1 −αt)(1 −βt)
where m is a small integer. In the second case, g
n
is
the sum of various terms, as many as there are terms
in the polynomial q(t), each one of the form:
q
k
[t
n−k
]
1

(1 −αt)(1 −βt)
.
It is therefore interesting to compute, once and for
all, the asymptotic value [t
n
]((1−αt)(1−βt))
s
, where
s = 1/2 or s = −1/2.
Let us suppose that [α[ > [β[, since the case α = β
has no interest and the case α = −β should be ap-
proached in another way. This hypothesis means that
t = 1/α is the radius of convergence of the function
and we can develop everything around this singular-
ity. In most combinatorial problems we have α > 0,
because the coefficients of f(t) are positive numbers,
but this is not a limiting factor.
Let us consider s = 1/2; in this case, a minus sign
should precede the square root. The evaluation is
shown in Table 7.1. The formula so obtained can be
considered sufficient for obtaining both the asymp-
totic evaluation of f
n
and a suitable numerical ap-
proximation. However, we can use the following de-
velopments:

2n
n

=
4
n
√
πn

1 −
1
8n
+
1
128n
2
+O

1
n
3

1
2n −1
=
1
2n

1 +
1
2n
+
1
4n
2
+O

1
n
3

1
2n −3
=
1
2n

1 +
3
2n
+O

1
n
2

and get:
f
n
=

α −β
α
α
n
2n
√
πn

1 −
6 + 3(α −β)
8(α −β)n
+
25
128n
2
+
+
9
8(α −β)n
2
−
9αβ + 51β
2
32(α −β)
2
n
2
+O

1
n
3

.
The reader is invited to find a similar formula for
the case s = 1/2.
7.8 Hayman’s method
The method for coefficient evaluation which uses the
function singularities (Darboux’ method), can be im-
proved and made more accurate, as we have seen,
by the technique of “subtracted singularities”. Un-
fortunately, these methods become useless when the
function f(t) has no singularity (entire functions) or
when the dominating singularity is essential. In fact,
in the former case we do not have any singularity to
operate on, and in the latter the development around
the singularity gives rise to a series with an infinite
number of terms of negative degree.
90 CHAPTER 7. ASYMPTOTICS
[t
n
] −(1 −αt)
1/2
(1 −βt)
1/2
= [t
n
] −(1 −αt)
1/2

α−β
α
+
β
α
(1 −αt)

1/2
=
= −

α−β
α
[t
n
](1 −αt)
1/2

1 +
β
α−β
(1 −αt)

1/2
=
= −

α−β
α
[t
n
](1 −αt)
1/2

1 +
β
2(α−β)
(1 −αt) −
β
2
8(α−β)
2
(1 −αt)
2
+

=
= −

α−β
α
[t
n
]

(1 −αt)
1/2
+
β
2(α−β)
(1 −αt)
3/2
−
β
2
8(α−β)
2
(1 −αt)
5/2
+

=
= −

α−β
α


1/2
n

(−α)
n
+
β
2(α−β)

3/2
n

(−α)
n
−
β
2
8(α−β)
2

5/2
n

(−α)
n
+

=
= −

α−β
α
(−1)
n−1
4
n
(2n−1)

2n
n

(−α)
n

1 −
β
2(α−β)
3
2n−3
−
β
2
8(α−β)
2
15
(2n−3)(2n−5)
+

=
=

α−β
α
α
n
4
n
(2n−1)

2n
n


1 −
3β
2(α−β)(2n−3)
−
15β
2
8(α−β)
2
(2n−3)(2n−5)
+O

1
n
3


.
Table 7.1: The case s = 1/2
In these cases, the only method seems to be the
Cauchy theorem, which allows us to evaluate [t
n
]f(t)
by means of an integral:
f
n
=
1
2πi

γ
f(t)
t
n+1
dt
where γ is a suitable path enclosing the origin. We do
not intend to develop this method here, but we’ll limit
ourselves to sketch a method, derived from Cauchy
theorem, which allows us to find an asymptotic evalu-
ation for f
n
in many practical situations. The method
can be implemented on a computer in the following
sense: given a function f(t), in an algorithmic way we
can check whether f(t) belongs to the class of func-
tions for which the method is applicable (the class
of “H-admissible” functions) and, if that is the case,
we can evaluate the principal value of the asymp-
totic estimate for f
n
. The system ΛΥΩ, by Flajolet,
Salvy and Zimmermann, realizes this method. The
development of the method was mainly performed
by Hayman and therefore it is known as Hayman’s
method; this also justifies the use of the letter H in
the definition of H-admissibility.
A function is called H-admissible if and only if it
belongs to one of the following classes or can be ob-
tained, in a finite number of steps according to the
following rules, from other H-admissible functions:
1. if f(t) and g(t) are H-admissible functions and
p(t) is a polynomial with real coefficients and
positive leading term, then:
exp(f(t)) f(t) +g(t) f(t) +p(t)
p(f(t)) p(t)f(t)
are all H-admissible functions;
2. if p(t) is a polynomial with positive coefficients
and not of the form p(t
k
) for k > 1, then the
function exp(p(t)) is H-admissible;
3. if α, β are positive real numbers and γ, δ are real
numbers, then the function:
f(t) = exp

β
(1 −t)
α

1
t
ln
1
(1 −t)

γ



2
t
ln

1
t
ln
1
(1 −t)

δ

is H-admissible.
For example, the following functions are all H-
admissible:
e
t
exp

t +
t
2
2

exp

t
1 −t

exp

1
t(1 −t)
2
ln
1
1 −t

.
In particular, for the third function we have:
exp

t
1 −t

= exp

1
1 −t
−1

=
1
e
exp

1
1 −t

and naturally a constant does not influence the H-
admissibility of a function. In this example we have
α = β = 1 and γ = δ = 0.
For H-admissible functions, the following result
holds:
Theorem 7.8.1 Let f(t) be an H-admissible func-
tion; then:
f
n
= [t
n
]f(t) ∼
f(r)
r
n

2πb(r)
as n → ∞
where r = r(n) is the least positive solution of the
equation tf
′
(t)/f(t) = n and b(t) is the function:
b(t) = t
d
dt

t
f
′
(t)
f(t)

.
As we said before, the proof of this theorem is
based on Cauchy’s theorem and is beyond the scope
of these notes. Instead, let us show some examples
to clarify the application of Hayman’s method.
7.9. EXAMPLES OF HAYMAN’S THEOREM 91
7.9 Examples of Hayman’s
Theorem
The first example can be easily verified. Let f(t) = e
t
be the exponential function, so that we know f
n
=
1/n!. For applying Hayman’s theorem, we have to
solve the equation te
t
/e
t
= n, which gives r = n.
The function b(t) is simply t and therefore we have:
[t
n
]e
t
=
e
n
n
n
√
2πn
and in this formula we immediately recognize Stirling
approximation for factorials.
Examples become early rather complex and require
a large amount of computations. Let us consider the
following sum:
n−1
¸
k=0

n −1
k

1
(k + 1)!
=
n
¸
k=0

n −1
k −1

1
k!
=
=
n
¸
k=0

n
k

−

n −1
k

1
k!
=
=
n
¸
k=0

n
k

1
k!
−
n−1
¸
k=0

n −1
k

1
k!
=
= [t
n
]
1
1 −t
exp

t
1 −t

−
−[t
n−1
]
1
1 −t
exp

t
1 −t

=
= [t
n
]
1 −t
1 −t
exp

t
1 −t

= [t
n
] exp

t
1 −t

.
We have already seen that this function is H-
admissible, and therefore we can try to evaluate the
asymptotic behavior of the sum. Let us define the
function g(t) = tf
′
(t)/f(t), which in the present case
is:
g(t) =
t
(1−t)
2
exp

t
1−t

exp

t
1−t
=
t
(1 −t)
2
.
The value of r is therefore given by the minimal pos-
itive solution of:
t
(1 −t)
2
= n or nt
2
−(2n + 1)t +n = 0.
Because ∆ = 4n
2
+ 4n + 1 − 4n
2
= 4n + 1, we have
the two solutions:
r =
2n + 1 ±
√
4n + 1
2n
and we must accept the one with the ‘−’ sign, which is
positive and less than the other. It is surely positive,
because
√
4n + 1 < 2
√
n + 1 < 2n, for every n > 1.
As n grows, we can also give an asymptotic approx-
imation of r, by developing the expression inside the
square root:
√
4n + 1 = 2
√
n

1 +
1
4n
=
= 2
√
n

1 +
1
8n
+O

1
n
2

.
From this formula we immediately obtain:
r = 1 −
1
√
n
+
1
2n
−
1
8n
√
n
+O

1
n
2

which will be used in the next approximations.
First we compute an approximation for f(r), that
is exp(r/(1 −r)). Since:
1 −r =
1
√
n
−
1
2n
+
1
8n
√
n
+O

1
n
2

=
=
1
√
n

1 −
1
2
√
n
+
1
8n
+O

1
n
√
n

we immediately obtain:
r
1 −r
=

1 −
1
√
n
+
1
2n
+O

1
n
√
n



√
n

1 +
1
2
√
n
+
1
8n
+O

1
n
√
n

=
=
√
n

1 −
1
2
√
n
+
1
8n
+O

1
n
√
n

=
=
√
n −
1
2
+
1
8
√
n
+O

1
n

.
Finally, the exponential gives:
exp

r
1 −r

=
e
√
n
√
e
exp

1
8
√
n
+O

1
n

=
=
e
√
n
√
e

1 +
1
8
√
n
+O

1
n

.
Because Hayman’s method only gives the principal
value of the result, the correction can be ignored (it
can be not precise) and we get:
exp

r
1 −r

=
e
√
n
√
e
.
The second part we have to develop is 1/r
n
, which
can be computed when we write it as exp(nln 1/r),
that is:
1
r
n
= exp

nln

1 +
1
√
n
+
1
2n
+O

1
n
√
n

=
= exp

n

1
√
n
+
1
2n
+O

1
n
√
n

−
−
1
2

1
√
n
+O

1
n
√
n

2
+O

1
n
√
n


92 CHAPTER 7. ASYMPTOTICS
= exp

n

1
√
n
+
1
2n
−
1
2n
+O

1
n
√
n

=
= exp

√
n +O

1
√
n

∼ e
√
n
.
Again, the correction is ignored and we only consider
the principal value. We now observe that f(r)/r
n
∼
e
2
√
n
/
√
e.
Only b(r) remains to be computed; we have b(t) =
tg
′
(t) where g(t) is as above, and therefore we have:
b(t) = t
d
dt
t
(1 −t)
2
=
t(1 +t)
(1 −t)
3
.
This quantity can be computed a piece at a time.
First:
r(1 +r) =

1 −
1
√
n
+
1
2n
+O

1
n
√
n




2 −
1
√
n
+
1
2n
+O

1
n
√
n

=
= 2 −
3
√
n
+
5
2n
+O

1
n
√
n

=
= 2

1 −
3
2
√
n
+
5
4n
+O

1
n
√
n

.
By using the expression already found for (1 −r) we
then have:
(1 −r)
3
=
1
n
√
n

1 −
1
2
√
n
+
1
8n
+O

1
n
√
n

3
=
=
1
n
√
n

1 −
1
√
n
+
1
2n
+O

1
n
√
n




1 −
1
2
√
n
+
1
8n
+O

1
n
√
n

=
=
1
n
√
n

1 −
3
2
√
n
+
9
8n
+O

1
n
√
n

.
By inverting this quantity, we eventually get:
b(r) =
r(1 +r)
(1 −r)
3
=
= 2n
√
n

1 +
3
2
√
n
+
9
8n
+O

1
n
√
n




1 −
3
2
√
n
+
5
4n
+O

1
n
√
n

=
= 2n
√
n

1 +
1
8n
+O

1
n
√
n

.
The principal value is 2n
√
n and therefore:

2πb(r) =

4πn
√
n = 2
√
πn
3/4
;
the final result is:
f
n
= [t
n
] exp

t
1 −t

∼
e
2
√
n
2
√
πen
3/4
.
It is a simple matter to execute the original sum
¸
n−1
k=0

n−1
k

/(k + 1)! by using a personal computer
for, say, n = 100, 200, 300. The results obtained can
be compared with the evaluation of the previous for-
mula. We can thus verify that the relative error de-
creases as n increases.
Chapter 8
Bibliography
The birth of Computer Science and the need of ana-
lyzing the behavior of algorithms and data structures
have given a strong twirl to Combinatorial Analysis,
and to the mathematical methods used to study com-
binatorial objects. So, near to the traditional litera-
ture on Combinatorics, a number of books and papers
have been produced, relating Computer Science and
the methods of Combinatorial Analysis. The first au-
thor who systematically worked in this direction was
surely Donald Knuth, who published in 1968 the first
volume of his monumental “The Art of Computer
Programming”, the first part of which is dedicated
to the mathematical methods used in Combinatorial
Analysis. Without this basic knowledge, there is little
hope to understand the developments of the analysis
of algorithms and data structures:
Donald E. Knuth: The Art of Computer Pro-
gramming: Fundamental Algorithms, Vol. I,
Addison-Wesley (1968).
Many additional concepts and techniques are also
contained in the third volume:
Donald E. Knuth: The Art of Computer Pro-
gramming: Sorting and Searching, Vol. III,
Addison-Wesley (1973).
Numerical and probabilistic developments are to
be found in the central volume:
Donald E. Knuth: The Art of Computer Pro-
gramming: Numerical Algorithms, Vol. II,
Addison-Wesley (1973)
A concise exposition of several important tech-
niques is given in:
Daniel H. Greene, Donald E. Knuth: Mathemat-
ics for the Analysis of Algorithms, Birkh¨auser
(1981).
The most systematic work in the field is perhaps
the following text, very clear and very comprehensive.
It deals, in an elementary but rigorous way, with the
main topics of Combinatorial Analysis, with an eye
to Computer Science. Many exercise (with solutions)
are proposed, and the reader is never left alone in
front of the many-faced problems he or she is encour-
aged to tackle.
Ronald L. Graham, Donald E. Knuth, Oren
Patashnik: Concrete Mathematics, Addison-
Wesley (1989).
Many texts on Combinatorial Analysis are worth of
being considered, because they contain information
on general concepts, both from a combinatorial and
a mathematical point of view:
William Feller: An Introduction to Probabil-
ity Theory and Its Applications, Wiley (1950) -
(1957) - (1968).
John Riordan: An Introduction to Combinatorial
Analysis, Wiley (1953) (1958).
Louis Comtet: Advanced Combinatorics, Reidel
(1974).
Ian P. Goulden, David M. Jackson: Combinato-
rial Enumeration, Dover Publ. (2004).
Richard P. Stanley: Enumerative Combinatorics,
Vol. I, Cambridge Univ. Press (1986) (2000).
Richard P. Stanley: Enumerative Combinatorics,
Vol. II, Cambridge Univ. Press !997) (2001).
Robert Sedgewick, Philippe Flajolet: An Intro-
duction to the Analysis of Algorithms, Addison-
Wesley (1995).
93
94 CHAPTER 8. BIBLIOGRAPHY
* * *
Chapter 1: Introduction
The concepts relating to the problem of searching
can be found in any text on Algorithms and Data
Structures, in particular Volume III of the quoted
“The Art of Computer Programming”. Volume I can
be consulted for Landau notation, even if it is com-
mon in Mathematics. For the mathematical concepts
of Γ and ψ functions, the reader is referred to:
Milton Abramowitz, Irene A. Stegun: Handbook
of Mathematical Functions, Dover Publ. (1972).
* * *
Chapter 2: Special numbers
The quoted book by Graham, Knuth and Patash-
nik is appropriate. Anyhow, the quantity we consider
are common to all parts of Mathematics, in partic-
ular of Combinatorial Analysis. The ζ function and
the Bernoulli numbers are also covered by the book of
Abramowitz and Stegun. Stanley dedicates to Cata-
lan numbers a special survey in his second volume.
Probably, they are the most frequently used quan-
tity in Combinatorics, just after binomial coefficients
and before Fibonacci numbers. Stirling numbers are
very important in Numerical Analysis, but here we
are more interested in their combinatorial meaning.
Harmonic numbers are the “discrete” version of log-
arithms, and from this fact their relevance in Combi-
natorics and in the Analysis of Algorithms originates.
Sequences studied in Combinatorial Analysis have
been collected and annotated by Sloane. The result-
ing book (and the corresponding Internet site) is one
of the most important reference point:
N. J. A. Sloane, Simon Plouffe: The Encyclope-
dia of Integer Sequences Academic Press (1995).
Available at:
http://www.research.att.com/ njas/sequences/.
* * *
Chapter 3: Formal Power Series
Formal power series have a long tradition; the
reader can find their algebraic foundation in the book:
Peter Henrici: Applied and Computational Com-
plex Analysis, Wiley (1974) Vol. I; (1977) Vol.
II; (1986) Vol. III.
Coefficient extraction is central in our approach; a
formalization can be found in:
Donatella Merlini, Renzo Sprugnol, M. Cecilia
Verri: The method of coefficients, The American
Mathematical Monthly (to appear).
The Lagrange Inversion Formula can be found in
most of the quoted texts, in particular in Stanley
and Henrici. Nowadays, almost every part of Math-
ematics is available by means of several Computer
Algebra Systems, implementing on a computer the
ideas described in this chapter, and many, many oth-
ers. Maple and Mathematica are among the most
used systems; other software is available, free or not.
These systems have become an essential tool for de-
veloping any new aspect of Mathematics.
* * *
Chapter 4: Generating functions
In our present approach the concept of an (ordi-
nary) generating function is essential, and this justi-
fies our insistence on the two operators “generating
function” and “coefficient of”. Since their introduc-
tion in Mathematics, due to de Moivre in the XVIII
Century, generating functions have been a controver-
sial concept. Now they are accepted almost univer-
sally, and their theory has been developed in many
of the quoted texts. In particular:
Herbert S. Wilf: Generatingfunctionology, Aca-
demic Press (1990).
A practical device has been realized in Maple:
Bruno Salvy, Paul Zimmermann: GFUN: a Maple
package for the manipulation of generating and
holonomic functions in one variable, INRIA, Rap-
port Technique N. 143 (1992).
The number of applications of generating functions
is almost infinite, so we limit our considerations to
some classical cases relative to Computer Science.
Sometimes, we intentionally complicate proofs in or-
der to stress the use of generating function as a uni-
fying approach. So, our proofs should be compared
with those given (for the same problems) by Knuth
or Flajolet and Sedgewick.
* * *
Chapter 5: Riordan arrays
95
Riordan arrays are part of the general method of
coefficients and are particularly important in prov-
ing combinatorial identities and generating function
transformations. They were introduced by:
Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan,
Leon C. Woodson: The Riordan group, Discrete
Applied Mathematics, 34 (1991) 229 – 239.
Their practical relevance was noted in:
Renzo Sprugnoli: Riordan arrays and combinato-
rial sums, Discrete Mathematics, 132 (1992) 267
– 290.
The theory was further developed in:
Donatella Merlini, Douglas G. Rogers, Renzo
Sprugnoli, M. Cecilia Verri: On some alterna-
tive characterizations of Riordan arrays, Cana-
dian Journal of Mathematics 49 (1997) 301 – 320.
Riordan arrays are strictly related to “convolution
matrices” and to “Umbral calculus”, although, rather
strangely, nobody seems to have noticed the connec-
tions of these concepts and combinatorial sums:
Steve Roman: The Umbral Calculus Academic
Press (1984).
Donald E. Knuth: Convolution polynomials, The
Methematica Journal, 2 (1991) 67 – 78.
Collections of combinatorial identities are:
Henry W. Gould: Combinatorial Identities. A
Standardized Set of Tables Listing 500 Binomial
Coefficient Summations West Virginia University
(1972).
Josef Kaucky: Combinatorial Identities Veda,
Bratislava (1975).
Other methods to prove combinatorial identities
are important in Combinatorial Analysis. We quote:
John Riordan: Combinatorial Identities, Wiley
(1968).
G. P. Egorychev: Integral Representation and the
Computation of Combinatorial Sums American
Math. Society Translations, Vol. 59 (1984).
Doron Zeilberger: A holonomic systems approach
to special functions identities, Journal of Compu-
tational and Applied Mathematics 32 (1990), 321
– 368.
Doron Zeilberger: A fast algorithm for proving
terminating hypergeometric identities, Discrete
Mathematics 80 (1990), 207 – 211.
M. Petkovˇsek, Herbert S. Wilf, Doron Zeilberger:
A = B, A. K. Peters (1996).
In particular, the method of Wilf and Zeilberger
completely solves the problem of combinatorial iden-
tities “with hypergeometric terms”. This means that
we can algorithmically establish whether an identity
is true or false, provided the two members of the iden-
tity:
¸
k
L(k, n) = R(n) have a special form:
L(k + 1, n)
L(k, n)
L(k, n + 1)
L(k, n)
R(n + 1)
R(n)
are all rational functions in n and k. This actu-
ally means that L(k, n) and R(n) are composed of
factorial, powers and binomial coefficients. In this
sense, Riordan arrays are less powerfy+ul, but can
be used also when non-hypergeometric terms are not
involved, as for examples in the case of harmonic and
Stirling numbers of both kind.
* * *
Chapter 6: Formal Methods
In this chapter we have considered two important
methods: the symbolic method for deducing counting
generating functions from the syntactic definition of
combinatorial objects, and the method of “operators”
for obtaining combinatorial identities from relations
between transformations of sequences defined by op-
erators.
The symbolic method was started by
Sch¨ utzenberger and Viennot, who devised a
technique to automatically generate counting gener-
ating functions from a context-free non-ambiguous
grammar. When the grammar defines a class of
combinatorial objects, this method gives a direct way
to obtain monovariate or multivariate generating
functions, which allow to solve many problems
relative to the given objects. Since context-free
languages only define algebraic generating functions
(the subset of regular grammars is limited to rational
96 CHAPTER 8. BIBLIOGRAPHY
functions), the method is not very general, but
is very effective whenever it can be applied. The
method was extended by Flajolet to some classes of
exponential generating functions and implemented
in Maple.
Marco Sch¨ utzenberger: Context-free languages
and pushdown automata, Information and Con-
trol 6 (1963) 246 – 264
Maylis Delest, Xavier Viennot: Algebraic lan-
guages and polyominoes enumeration, X Collo-
quium on Automata, Languages and Program-
ming - Lecture Notes in Computer Science (1983)
173 – 181.
Philippe Flajolet: Symbolic enumerative com-
binatorics and complex asymptotic analysis,
Algorithms Seminar, (2001). Available at:
http://algo.inria.fr/seminars/sem00-
01/flajolet.html
The method of operators is very old and was devel-
oped in the XIX Century by English mathematicians,
especially George Boole. A classical book in this di-
rection is:
Charles Jordan: Calculus of Finite Differences,
Chelsea Publ. (1965).
Actually, the method is used in Numerical Anal-
ysis, but it has a clear connection with Combinato-
rial Analysis, as our numerous examples show. The
important concepts of indefinite and definite summa-
tions are used by Wilf and Zeilberger in the quoted
texts. The Euler-McLaurin summation formula is the
first connection between finite methods (considered
up to this moment) and asymptotics.
* * *
Chapter 7: Asymptotics
The methods treated in the previous chapters are
“exact”, in the sense that every time they give the so-
lution to a problem, this solution is a precise formula.
This, however, is not always possible, and many times
we are not able to find a solution of this kind. In these
cases, we would also be content with an approximate
solution, provided we can give an upper bound to the
error committed. The purpose of asymptotic meth-
ods is just that.
The natural settings for these problems are Com-
plex Analysis and the theory of series. We have used
a rather descriptive approach, limiting our consid-
erations to elementary cases. These situations are
covered by the quoted text, especially Knuth, Wilf
and Henrici. The method of Heyman is based on the
paper:
Micha Hofri: Probabilistic Analysis of Algo-
rithms, Springer (1987).
Index
Γ function, 8, 84
ΛΥΩ, 90
ψ function, 8
1-1 correspondence, 11
1-1 mapping, 11
A-sequence, 58
absolute scale, 9
addition operator, 77
Adelson-Velski, 52
adicity, 35
algebraic singularity, 86, 87
algebraic square root, 87
Algol’60, 69
Algol’68, 70
algorithm, 5
alphabet, 12, 67
alternating subgroup, 14
ambiguous grammar, 68
Appell subgroup, 57
arithmetic square root, 87
arrangement, 11
asymptotic approximation, 88
asymptotic development, 80
asymptotic evaluation, 88
average case analysis, 5
AVL tree, 52
Backus Normal Form, 70
Bell number, 24
Bell subgroup, 57
Bender’s theorem, 84
Bernoulli number, 18, 24, 53, 80, 85
big-oh notation, 9
bijection, 11
binary searching, 6
binary tree, 20
binomial coefficient, 8, 15
binomial formula, 15
bisection formula, 41
bivariate generating function, 55
BNF, 70
Boole, George, 73
branch, 87
C language, 69
cardinality, 11
Cartesian product, 11
Catalan number, 20, 34
Catalan triangle, 63
Cauchy product, 26
Cauchy theorem, 90
central binomial coefficient, 16, 17
central trinomial coefficient, 46
characteristic function, 12
Chomski, Noam, 67
choose, 15
closed form expression, 7
codomain, 11
coefficient extraction rules, 30
coefficient of operator, 30
coefficient operator, 30
colored walk, 62
colored walk problem, 62
column, 11
column index, 11
combination, 15
combination with repetitions, 16
complete colored walk, 62
composition of f.p.s., 29
composition of permutations, 13
composition rule for coefficient of, 30
composition rule for generating functions, 40
compositional inverse of a f.p.s., 29
Computer Algebra System, 34
context free grammar, 68
context free language, 68
convergence, 83
convolution, 26
convolution rule for coefficient of, 30
convolution rule for generating functions, 40
cross product rule, 17
cycle, 12, 21
cycle degree, 12
cycle representation, 12
Darboux’ method, 84
definite integration of a f.p.s., 28
definite summation, 78
degree of a permutation, 12
delta series, 29
derangement, 12, 86
derivation, 67
diagonal step, 62
97
98 INDEX
diagonalisation rule for generating functions, 40
difference operator, 73
differentiation of a f.p.s., 28
differentiation rule for coefficient of, 30
differentiation rule for generating functions, 40
digamma function, 8
disposition, 15
divergence, 83
domain, 11
dominating singularity, 85
double sequence, 11
Dyck grammar, 68
Dyck language, 68
Dyck walk, 21
Dyck word, 69
east step, 20, 62
empty word, 12, 67
entire function, 89
essential singularity, 86
Euclid’s algorithm, 19
Euler constant, 8, 18
Euler transformation, 42, 55
Euler-McLaurin summation formula, 80
even permutation, 13
exponential algorithm, 10
exponential generating function, 25, 39
exponentiation of f.p.s., 28
extensional definition, 11
extraction of the coefficient, 29
f.p.s., 25
factorial, 8, 14
falling factorial, 15
Fibonacci number, 19
Fibonacci problem, 19
Fibonacci word, 70
Fibonacci, Leonardo, 18
finite operator, 73
fixed point, 12
Flajolet, Philippe, 90
formal grammar, 67
formal language, 67
formal Laurent series, 25, 27
formal power series, 25
free monoid, 67
full history recurrence, 47
function, 11
Gauss’ integral, 8
generalized convolution rule, 56
generalized harmonic number, 18
generating function, 25
generating function rules, 40
generation, 67
geometric series, 30
grammar, 67
group, 67
H-admissible function, 90
Hardy’s identity, 61
harmonic number, 8, 18
harmonic series, 17
Hayman’s method, 90
head, 67
height balanced binary tree, 52
height of a tree, 52
i.p.l., 51
identity, 12
identity for composition, 29
identity operator, 73
identity permutation, 13
image, 11
inclusion exclusion principle, 86
indefinite precision, 35
indefinite summation, 78
indeterminate, 25
index, 11
infinitesimal operator, 73
initial condition, 6, 47
injective function, 11
input, 5
integral lattice, 20
integration of a f.p.s., 28
intensional definition, 11
internal path length, 51
intractable algorithm, 10
intrinsecally ambiguous grammar, 69
invertible f.p.s., 26
involution, 13, 49
juxtaposition, 67
k-combination, 15
key, 5
Kronecker’s delta, 43
Lagrange, 32
Lagrange inversion formula, 33
Lagrange subgroup, 57
Landau, Edmund, 9
Landis, 52
language, 12, 67
language generated by the grammar, 68
leftmost occurrence, 67
length of a walk, 62
length of a word, 67
letter, 12, 67
LIF, 33
linear algorithm, 10
linear recurrence, 47
linear recurrence with constant coefficients, 48
INDEX 99
linear recurrence with polynomial coefficients, 48
linearity rule for coefficient of, 30
linearity rule for generating functions, 40
list representation, 36
logarithm of a f.p.s., 28
logarithmic algorithm, 10
logarithmic singularity, 88
mapping, 11
Mascheroni constant, 8, 18
metasymbol, 70
method of shifting, 44
Miller, J. C. P., 37
monoid, 67
Motzkin number, 71
Motzkin triangle, 63
Motzkin word, 71
multiset, 24
negation rule, 16
Newton’s rule, 28, 30, 84
non convergence, 83
non-ambiguous grammar, 68
north step, 20, 62
north-east step, 20
number of involutions, 14
number of mappings, 11
Numerical Analysis, 73
O-notation, 9
object grammar, 71
occurence, 67
occurs in, 67
odd permutation, 13
operand, 35
operations on rational numbers, 35
operator, 35, 72
order of a f.p.s., 25
order of a pole, 85
order of a recurrence, 47
ordered Bell number, 24, 85
ordered partition, 24
ordinary generating function, 25
output, 5
p-ary tree, 34
parenthetization, 20, 68
partial fraction expansion, 30, 48
partial history recurrence, 47
partially recursive set, 68
Pascal language, 69
Pascal triangle, 16
path, 20
permutation, 12
place marker, 25
Pochhammer symbol, 15
pole, 85
polynomial algorithm, 10
power of a f.p.s., 27
preferential arrangement number, 24
preferential arrangements, 24
prefix, 67
principal value, 88
principle of identity, 40
problem of searching, 5
product of f.L.s., 27
product of f.p.s., 26
production, 67
program, 5
proper Riordan array, 55
quadratic algoritm, 10
quasi-unit, 29
rabbit problem, 19
radius of convergence, 83
random permutation, 14
range, 11
recurrence relation, 6, 47
renewal array, 57
residue of a f.L.s., 30
reverse of a f.p.s., 29
Riemann zeta function, 18
Riordan array, 55
Riordan’s old identity, 61
rising factorial, 15
Rogers, Douglas, 57
root, 21
rooted planar tree, 21
row, 11
row index, 11
row-by-column product, 32
Salvy, Bruno, 90
Sch¨ utzenberger methodology, 68
semi-closed form, 46
sequence, 11
sequential searching, 5
set, 11
set partition, 23
shift operator, 73
shifting rule for coefficient of, 30
shifting rule for generating functions, 40
shuffling, 14
simple binomial coefficients, 59
singularity, 85
small-oh notation, 9
solving a recurrence, 6
sorting, 12
south-east step, 20
square root of a f.p.s., 28
Stanley, 33
100 INDEX
Stirling number of the first kind, 21
Stirling number of the second kind, 22
Stirling polynomial, 23
Stirling, James, 21, 22
subgroup of associated operators, 57
subtracted singularity, 88
subword, 67
successful search, 5
suffix, 67
sum of a geometric progression, 44
sum of f.L.s, 27
sum of f.p.s., 26
summation by parts, 78
summing factor method, 49
surjective function, 11
symbol, 12, 67
symbolic method, 68
symmetric colored walk problem, 62
symmetric group, 14
symmetry formula, 16
syntax directed compilation, 70
table, 5
tail, 67
Tartaglia triangle, 16
Taylor theorem, 80
terminal word, 68
tractable algorithm, 10
transposition, 12
transposition representation, 13
tree representation, 36
triangle, 55
trinomial coefficient, 46
unary-binary tree, 71
underdiagonal colored walk, 62
underdiagonal walk, 20
unfold a recurrence, 6, 49
uniform convergence, 83
unit, 26
unsuccessful search, 5
van Wijngarden grammar, 70
Vandermonde convolution, 43
vector notation, 11
vector representation, 12
walk, 20
word, 12, 67
worst AVL tree, 52
worst case analysis, 5
Z-sequence, 58
zeta function, 18
Zimmermann, Paul, 90