This book is of interest to mathematicians and computer scientists working in finite mathematics and combinatorics. It presents a breakthrough method for analyzing complex summations. Beautifully written, the book contains practical applications as well as conceptual developments that will have applications in other areas of mathematics.
From the table of contents: * Proof Machines * Tightening the Target * The Hypergeometric Database * The Five Basic Algorithms: Sister Celine's Method, Gosper&'s Algorithm, Zeilberger's Algorithm, The WZ Phenomenon, Algorithm Hyper * Epilogue: An Operator Algebra Viewpoint * The WWW Sites and the Software (Maple and Mathematica) Each chapter contains an introduction to the subject and ends with a set of exercises.
Author(s): Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
Edition: 1
Publisher: A K Peters/CRC Press
Year: 1996
Language: English
Pages: 224
Cover
Title Page
Table of Contents
Foreword
A Quick Start
I Background
1 Proof Machines
1.1 Evolution of the province of human thought
1.2 Canonical and normal forms
1.3 Polynomial identities
1.4 Proofs by example?
1.5 Trigonometric identities
1.6 Fibonacci identities
1.7 Symmetric function identities
1.8 Elliptic function identities
2 Tightening the Target
2.1 Introduction
2.2 Identities
2.3 Human and computer proofs; an example
2.4 A Mathematica session
2.5 A Maple session
2.6 Where we are and what happens next
2.7 Exercises
3 The Hypergeometric Database
3.1 Introduction
3.2 Hypergeometric series
3.3 How to identify a series as hypergeometric
3.4 Software that identifies hypergeometric series
3.5 Some entries in the hypergeometric database
3.6 Using the database
3.7 Is there really a hypergeometric database?
3.8 Exercises
II The Five Basic Algorithms
4 Sister Celine’s Method
4.1 Introduction
4.2 Sister Mary Celine Fasenmyer
4.3 Sister Celine’s general algorithm
4.4 The Fundamental Theorem
4.5 Multivariate and “q” generalizations
4.6 Exercises
5 Gosper’s Algorithm
5.1 Introduction
5.2 Hypergeometrics to rationals to polynomials
5.3 The full algorithm: Step 2
5.4 The full algorithm: Step 3
5.5 More examples
5.6 Similarity among hypergeometric terms
5.7 Exercises
6 Zeilberger’s Algorithm
6.1 Introduction
6.2 Existence of the telescoped recurrence
6.3 How the algorithm works
6.4 Examples
6.5 Use of the programs
6.6 Exercises
7 The WZ Phenomenon
7.1 Introduction
7.2 WZ proofs of the hypergeometric database
7.3 Spinoffs from the WZ method
7.4 Discovering new hypergeometric identities
7.5 Software for the WZ method
7.6 Exercises
8 Algorithm Hyper
8.1 Introduction
8.2 The ring of sequences
8.3 Polynomial solutions
8.4 Hypergeometric solutions
8.5 A Mathematica session
8.6 Finding all hypergeometric solutions
8.7 Finding all closed form solutions
8.8 Some famous sequences that do not have closed form
8.9 Inhomogeneous recurrences
8.10 Factorization of operators
8.11 Exercises
III Epilogue
9 An Operator Algebra Viewpoint
9.1 Early history
9.2 Linear difference operators
9.3 Elimination in two variables
9.4 Modified elimination problem
9.5 Discrete holonomic functions
9.6 Elimination in the ring of operators
9.7 Beyond the holonomic paradigm
9.8 Bi-basic equations
9.9 Creative anti-symmetrizing
9.10 Wavelets
9.11 Abel-type identities
9.12 Another semi-holonomic identity
9.13 The art
9.14 Exercises
A The WWW sites and the software
A.1 The Maple packages EKHAD and qEKHAD
A.2 Mathematica programs
Bibliography
Index
