# A = B

###### Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger
Publisher:
A K Peters/CRC Press
Publication Date:
1996
Number of Pages:
224
Format:
Hardcover
Price:
59.36
ISBN:
9781568810638
Category:
Monograph
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
04/23/2015
]

This is an exposition of a group of methods for working with hypergeometric sums, developed roughly from 1945 to 1995. Hypergeometric has a specific meaning here, but you can think of it roughly as any sum or infinite series where the summand is made up of binomial coefficients, factorials, polynomials, and powers. To get a quick overview of the power of these methods, look at the American Mathematical Monthly article “How To Do Monthly Problems With Your Computer” by Nemes et al. (v. 104, 1997, pp. 505–519), that gives a sample of 27 problems from the Monthly Problem Section that can be done mechanically using these methods.

The cryptic title A=B refers to the methods’ application to proving hypergeometric identities, but in most cases they can produce a closed form expression for a finite or infinite sum without your needing to guess it first. The book gives a gradual and mostly historical introduction to the algorithms: Sister Celine’s (most of historical interest today, but a good introduction), Gosper’s, Zeilberger’s creative telescoping, Wilf-Zeilberger (WZ), and a more powerful form of WZ called Hyper. The last chapter collects a lot of fragments and ideas for further work. The algorithms are all conceptually simple but messy to work out, so they are ideal for computer implementation, and some of the implementations (mostly in Maple and Mathematica) are discussed here.

The present book is nearly 20 years old. The algorithms given here are essentially the last word for the problems they can solve, so the book is still up-to-date as far as it goes. There have been some new algorithms developed since then. A more recent book (that I have not seen) is Koepf’s Hypergeometric Summation (2nd edition, 2014) that covers the same topics as the present book plus the newer algorithms; it is slanted towards implementation while the present book is slanted toward the mathematical theory.

Note that this book is also available online.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

• 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 formsolutions
• 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