Home Page for the Book “A=B”

About the Book

“A=B” is about identities in general, and hypergeometric identities in particular, with emphasis on computer methods of discovery and proof. The book describes a number of algorithms for doing these tasks, and we intend to maintain the latest versions of the programs that carry out these algorithms on this page. So be sure to consult this page from time to time, and help yourself to the latest versions of the programs.

 In addition to programs, we will post here other items of interest relating to the book, such as the current errata sheet (see below). The other side of the coin is that we invite your comments about the content of the book, the programs, any errors that you may discover, or whatever. You can send us your comments by e-mail if you wish.

The book is a selection of the Library of Science.

A Japanese translation of A=B, by Toppan Co., Ltd., appeared in November of 1997.

What’s new:

  • The version of EKHAD of February, 1999 has a helpful idea from Frederic Chyzak, which has resulted in a substantial speedup. Get your copy here.
  • We have a department of “Case Studies,” on this web site. We started it off with one or two short contributions of our own, but we invite all readers and users of these algorithms to send us nice writeups of interesting things that they have done with the computerized methods in A=B, and we’ll post them here.
  • Some university courses have been designed around this subject. Glenn Tesler taught such a course at UCSD, and has created a web site that contains a good bit of helpful material, software, homework problems, etc.

From the reviews …

  • “… If the complete automation of a major industry within discrete mathematics with relevance to computer science counts as the first miracle, this entertaining accessible exposition by the discoverers themselves counts as the second. … Seldom do we find such a dramatic mathematical breakthrough placed within the reach of such a large audience so soon. Highly recommended. Undergraduates through faculty.” – D. V. Feldman, University of New Hampshire [CHOICE, 34, Nov. 1996]
  • “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. [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 …” – D. M. Bressoud, Macalester College [Zentralblatt für Mathematik 848 (1996).]
  • “This marvellous expository text describes the authors’ answer to Exercise in a book by Knuth … All [the] building blocks are brilliantly discussed in the text which is written in a spirit that puts strong emphasis on tutorial rather than on research exposition aspects — a wise choice in view of the novelty and broad applicability of the material… the authors not only skillfully explained their methods to a computer, they also did a truly outstanding job in explaining these recent developments to a broad audience ranging from students to researchers. In particular, this book is a must for all those who at least once have struggled with a binomial sum.” – Peter Paule, RISC-Linz; Austria [Math. Revs. 97j:05001] Here is the complete text of this review.
  • This review of A=B, by Noam Zeilberger, nephew of author Doron Zeilberger, although it is from a possibly not impartial source, is in fact completely objective:       
    A=B by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger         What are you waiting for? Buy the book      Written in a wonderful expository style, this books succeeds in making its      difficult subject matter accessible to a wide variety of people.      Of course, mathematicians studying hypergeometric series will have great use      for this book.  However, non-mathematicians can also greatly benefit from      reading it.  Computer scientists will be interested in the authors’ unique      approach towards automated proofs.  A=B is enjoyable reading and so really      anyone with some desire to learn something about the field of      computer-generated proofs should get this book.  Above all, the book is a      great example of mathematical exposition and should be used as a standard      by those wishing to present their research to a large audience.        
    By the way, why don’t you visit the A=B page?
  • In the November-December, 1997 issue of The American Scientist, which is the organ of the honorary society Sigma Xi, you can read an article about our work, entitled `Gorilla’ Tackles Monster Sums.
  • “This remarkable book tells of a revolution akin to the one in symbolic integration nearly three decades ago. Until recently, combinatorial identities had to be proved by some clever argument, say by finding an appropriate bijection. Now computers have taken over. Three of the experts who enabled this automation have combined to produce a very readable account of their work … Throw out your catalogue of identities, … and buy `A=B’.” – Ian Wanless, Australian National University [Australian Math. Soc. Gazette,25, No. 1, April, 1998, 26-27.] Here is the full text of this review in Acrobat format.
  • “This remarkable book is extremely well-written and gives a complete self contained exposition of a fascinating breakthrough in the field of computer algebra and automatic theorem proving. It’s about computer programs for simplifying sums that involve binomial coefficients and for discovery and proof of hypergeometric identities. The authors of the book played key roles in these exciting new developments… The book is written in an exceptionally clear way and can be read by anyone who has had at least one year of university mathematics. The many examples, all verifiable in real time on your PC, make the book very lively. The material is absolutely fascinating, both for undergraduates and for professional mathematicians…” — Jan Denef, in J. Approximation Theory, October, 1999. You can read the full review here.
  • “The good news is that after 30 years of extraordinary efforts (largely the efforts of this book’s authors), this problem [developing computer programs to simplify hyergeometric sums] is largely solved … This book is about the problem, the history of its solution, the resulting algorithms, and finally — about the programs… Although the book is very technical, it is written in a very popular and understandable way. It starts with the basics, it gently guides the reader through the programs, through the formulas and through the numerous examples… I just love this book, and I hope y’all will too.”– Vladik Kreinovich, in SIGACT News31, No. 4, 2000. You can read the full review here.

Thanks to

  • Helmut Prodinger for finding a rarely occurring bug in EKHAD,

and to the following readers who have made thoughtful contributions to our errata sheet:

  • Joris Van der Jeugt and Griet Boterbergh
  • Laurent Habsieger
  • Richard E. Stone
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