Three of the foremost authorities in the field of discrete mathematics – Marko Petkovsek (University of Ljubljana), Herbert Wilf (University of Pennsylvania) and Doron Zeilberger (Temple University) have joined forced to create a discrete mathematics textbook. They discuss how computing technology has impacted the art of mathematics and proofs. According to one review of this book, the authors “have been at the forefront of a group of researchers who have found and implemented algorithmic approaches to the study of identities for hypergeometric and basic hypergeometric series.”

From another review posted on their website, “This book is an essential resource for anyone who ever encounters binomial coefficient identities, for anyone who is interested in how computers are being used to discover and prove mathematical identities and for anyone who simply enjoys a well-written book that presents interesting cutting edge mathematics in an accessible style.”

## Table of Contents

- 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
- 3. The Hypergeometric Database
- 3.1 Introduction
- 3.2 Hyper geometric 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?
- 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
- 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
- 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
- 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
- 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
- 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
- A. The WWW sites and the software
- A.1 The Maple packages EKHAD and qEKHAD
- A.2 Mathematica programs

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