Home Page for the Book "A=B"
with a Foreword by Donald E. Knuth
YOU CAN NOW
DOWNLOAD THE ENTIRE
BOOK !!
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 email
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 wellwritten 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
1.2.6.63 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, RISCLinz; 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, nonmathematicians 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
computergenerated 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 NovemberDecember, 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" (avaliable from Proquest)

"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, 2627.] Here is
the full text of this review in Acrobat format.

"This remarkable book is extremely wellwritten 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 News, 31, 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