Table Of Content~ SpringerWienNewYork
Texts and Monographs in Symbolic
Computation
A Series of the Research Institute
for Symbolic Computation,
Johannes Kepler University Linz, Austria
Series Editors
Robert Corless; University of Western Ontario, Canada
Hoon Hong; North Carolina State University, USA
Tetsuo Ida; University of Tsukuba, Japan
Martin Kreuzer; Universitat Passau, Germany
Bruno Salvy; INRIA Rocquencourt, France
Dongming Wang; Universite Pierre et Marie Curie - CNRS, France
Peter Paule; Johannes Kepler University Linz, Austria
For further volumes:
http://www.springer.com/series/3073
Manuel Kauers . Peter Paule
The Concrete Tetrahedron
Symbolic Sums, Recurrence Equations,
Generating Functions, Asymptotic Estimates
SpringerWienNewYork
Manuel Kauers Peter Paule
Research Institute Research Institute
for Symbolic Computation (RISC) for Symbolic Computation (RISC)
Johannes Kepler University Linz Johannes Kepler University Linz
4040 Linz, Austria 4040 Linz, Austria
[email protected] peter. [email protected]
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ISBN 978-3-709\-0444-6 SpringerWienNewYork
Preface
There are problems in mathematics which are so hard that they remain open for cen
turies. But many problems are not of this type. Often, being able to solve a problem
just depends on knowing the right technique. This book is a book on techniques.
More precisely, it is a book on techniques for solving problems about infinite se
quences. Some of these techniques have belonged already to the repertoire of Euler
and Gauss, others have been invented only a couple of years ago and are best suited
for being executed in a computer algebra system.
A course on such techniques has been taught by the second author for almost twenty
years at the Johannes Kepler University in Linz, Austria. The material of this course
has served as a starting point in writing the present book. Besides the teaching ex
perience with that course, also our personal experience with applying, developing,
and implementing algorithms for combinatorial sequences and special functions has
greatly influenced the text. The techniques we present have often worked for us, and
we are convinced that they will also work for our readers.
We have several goals. The first is to give an overview of some of the most im
portant mathematical questions about infinite sequences which can be answered by
the computer. It may seem at first glance that a user manual of a computer alge
bra system could serve the same purpose. But this is not quite so, because a proper
interpretation of a computational result may well require some knowledge of the
underlying theory, and sometimes a problem may be solvable only by some varia
tion of a classical algorithm. Our second goal is therefore also to explain how the
algorithms work and on what principles they are based. Our third goal, finally, is
to describe also some techniques which are suitable for traditional paper and pencil
reasoning rather than for modern computer calculations. This is still useful because
paper and pencil arguments are often needed to bring the problem at hand into an
equivalent form which then can be handled by a computer.
We have included more than one hundred problems on which the techniques ex
plained in the text can be tested. Some of the problems are meant to be solved on
paper, others by using computer algebra. The reader will also find problems which
only after some prior hand calculation can be completed electronically. These ones
vi
may consume some more time and thought than the others, but they surely offer the
best training effect, because these problems give the most accurate impression of
how the various techniques are applied in real life. Depending on the individual in
terests of the reader and on the particular problem at hand, it may be instructive not
to resort to the built-in commands of a computer algebra system for, say, evaluating
a certain sum in closed form, but instead to execute the algorithms described in the
text in a step-by-step manner, using the computer algebra system only for basic op
erations such as adding, multiplying, or factoring polynomials, or for solving linear
systems. We leave the choice of appropriate tools to the reader. But in either case,
we want to stress explicitly that the use of computers is not just allowed but also
strongly encouraged - not only for solving the problems in this book, but in general.
In order to spread this message as far as possible and to reach not only those who
already know, but also undergraduate mathematics or computer science students and
researchers from other mathematical or even non-mathematical disciplines, we have
preferred concrete examples over abstract theorems, and informal explanations over
formal proofs wherever this seemed to be in the interest of readability. We have
also strived for reducing the assumed background knowledge as much as possible.
The only knowledge which we will assume is some familiarity with basic algebraic
notions (rings, fields, etc.), with linear algebra in finite dimensional vector spaces,
and with some complex analysis in one variable. But not more.
With this background it is already possible to cover quite some material, including
some of the most spectacular discoveries in the area, which have been made in
the last decade of the 20th century. It is not our goal, however, to reach subjects
of ongoing research. Important topics such as summation with IT}: fields, special
function inequalities, holonomic functions in several variables, or objects defined
by nonlinear recurrence or differential equations are not addressed at all. But even
with this restriction, we do believe that the tools presented in this book are useful.
Let us finally take the opportunity to thank Victor Moll, Veronika Pillwein, and
Carsten Schneider for valuable comments and enlighting discussions. We especially
want to express our gratitude to Michael Singer and his students for carefully going
through a preliminary version of the book. Their remarks and suggestions have been
of great help in fixing errors and resolving confusion.
Hagenberg, October 2010 Manuel Kauers
Peter Paule
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Selection Sort and Quicksort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
1.2 Recurrence Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Symbolic Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Asymptotic Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
1.6 The Concrete Tetrahedron ............................. . . . . .. 13
1.7 Problems................................................. 15
2 Formal Power Series ........................................... 17
2.1 Basic Facts and Definitions .................................. 17
2.2 Differentiation and Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
2.3 Sequences of Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24
2.4 The Transfer Principle ...................................... 29
2.5 Multivariate Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
2.6 Truncated Power Series ..................................... 35
2.7 Problems................................................. 38
3 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
3.1 Polynomials as Power Series ................................. 43
3.2 Polynomials as Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
3.3 The Tetrahedron for Polynomials ............................. 48
3.4 Polynomials as Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
3.5 Polynomials as Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
3.6 Applications............................................... 55
3.7 Problems................................................. 58
viii Contents
4 C-Finite Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
4.1 Fibonacci Numbers ......................................... 63
4.2 Recurrences with Constant Coefficients. . . . . . . . . . . . . . . . . . . . . . .. 66
4.3 Closure Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
4.4 The Tetrahedron for C-finite Sequences. . . . . . . . . . . . . . . . . . . . . . .. 74
4.5 Systems of C-finite Recurrences. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78
4.6 Applications............................................... 80
4.7 Problems................................................. 84
5 Hypergeometric Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
5.1 The Binomial Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
5.2 Basic Facts and Definitions .................................. 90
5.3 The Tetrahedron for Hypergeometric Sequences. . . . . . . . . . . . . . . .. 94
5.4 Indefinite Summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
5.5 Definite Summation ........................................ 103
5.6 Applications ............................................... 107
5.7 Problems ................................................. 110
6 Algebraic Functions ............................................ 113
6.1 Catalan Numbers ........................................... 113
6.2 Basic Facts and Definitions .................................. 116
6.3 Puiseux Series and the Newton Polygon ........................ ll9
6.4 Closure Properties .......................................... 123
6.5 The Tetrahedron for Algebraic Functions ....................... 125
6.6 Applications ............................................... 130
6.7 Problems ................................................. 133
7 Holonomic Sequences and Power Series .......................... 137
7.1 Harmonic Numbers ......................................... 137
7.2 Equations with Polynomial Coefficients ........................ 139
7.3 Generalized Series Solutions ................................. 144
7.4 Closed Form Solutions ...................................... 148
7.5 The Tetrahedron for Holonomic Functions ...................... 152
7.6 Applications ............................................... 157
7.7 Problems ................................................. 160
Contents ix
Appendix .......................................................... 165
A.l Basic Notions and Notations ................................. 165
A.2 Basic Facts from Computer Algebra ........................... 167
A.3 A Collection of Formal Power Series Identities .................. 168
AA Closure Properties at One Glance ............................. 169
A.5 Software .................................................. 171
A.6 Solutions to Selected Problems ............................... 174
A.7 Bibliographic Remarks ...................................... 189
References ......................................................... 193
Subject Index ...................................................... 197
Chapter 1
Introduction
This is the story about four mathematical concepts which playa great role in many
different areas of mathematics. It is a story about symbolic sums, recurrence (dif
ference) equations, generating functions, and asymptotic estimates. In this book, we
will study their key features in isolation or in combination, their mastery by paper
and pencil or by computer programs, and their application to problems in pure math
ematics or to "real world problems". To begin with, we take a look at a particular
"real world problem" and see how sums, recurrence equations, generating functions,
and asymptotic estimates may naturally arise in such a context. After having intro
duced these four main characters of this book informally by seeing them in action,
we will then prepare the stage for their detailed study.
1.1 Selection Sort and Quicksort
Suppose you want to write a computer program that sorts a given array of numbers.
There are many ways to do this. Which one should you choose if you want your
program to be fast? The speed of a program can be estimated by counting the number
of operations it performs in dependence of the input size. For a sorting algorithm, it
is thus natural to ask how many comparisons of two elements are needed in order to
bring them into the right order.
The Selection Sort algorithm sorts a given array (aj, ... , an) of numbers as follows.
First it goes through the entire list and determines the minimum. Say ak is the min
imum. Then it exchanges aj and ak so that the minimum is at the right position.
Then it goes through the list again, starting at the second position, and searches
for the minimum of the remaining elements. It is exchanged with a2 so that it is
also at the right position, and so on. An example with eight numbers is shown in
Fig. 1.1.
M. Kauers, P. Paule, The Concrete Tetrahedron
© Springer-Verlag/Wien 2011
