23 of 24 people found the following review helpful:
5.0 out of 5 stars
Prize winning work, June 22, 2000
This review is from: A = B (Hardcover)
The mathematics described in the book "A=B" lead to the awarding of a Steele Prize by the American Mathematical Society to Herbert Wilf and Doron Zeilberger. What the authors have done is to discover a way of using computer algebra and some mathematical ideas they and others had earlier to make it possible to find the sum of most of the hypergeometric and basic hypergeometric series which can be summed. This method also leads to recurrence relations when polynomial series cannot be summed. It does not work for all series, but most of the time it does work. This book and the ideas in it should be part of the working knowledge of anyone who uses special functions of hypergeometric or basic hypergeometric type.
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18 of 19 people found the following review helpful:
5.0 out of 5 stars
Witty + Concise + Useful = Excellent, July 24, 2003
This review is from: A = B (Hardcover)
This book is a refreshingly witty and concise look at--of all things--hypergeometric identities. If you have never heard of a binomial coefficient, or shudder at the thought of a double factorial, of fail to see what's so special about special functions, then you are unlikely to derive much pleasure from this volume. On the other hand, if you've encountered gamma functions, Catalan numbers, and Grobner bases, and would delight in discovering deep connections among such vastly different ideas, or if you have wondered how computers can be used to discover or verify monstrous combinatorial identities, then you've stumbled upon a book that is sure to become a favorite.
This book introduces the idea of hypergeometric function, the Swiss army knife of combinatorial mathematics, and proceeds to develop algorithms for their computation as well as numerous applications. The authors also reveal what, exactly, computers can help us to decide, what is a "closed form" solution, what are "canonical" and "normal" forms, and inject relevant philosophical digressions that keep the discussions lively and entertaining. The authors also present snippets of "Mathematica" code so that you can try out many of the basic operations yourself.
The book has concise chapters on Sister Celine's method, Gosper's algorithm, Zeilberger's algorithm, and the WZ algorithm, with sufficient detail that you will likely be able to apply the algorithms yourself (perhaps by downloading the Mathematica packages that the authors point you to). The techniques are invaluable in proving identities in combinatorial mathematics; that is, identities involving binomial coefficients, factorials, rational functions, etc. By means of such techniques, computers "not only find proofs of known identities, they also find completely new identities. Lots of them. Some very pretty. Some not so pretty but very useful. Some neither pretty nor useful, in which case we humans can ignore them."
This is a well-written and highly accessible book about an important (albeit very narrow and specialized) branch of mathematics. If you have little experience with hypergeometric functions, yet deal with combinatorial mathematics, you will likely read this book in one (long) sitting; and you'll be glad you did.
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8 of 8 people found the following review helpful:
5.0 out of 5 stars
A delightful read for any mathematician!, November 15, 2006
This review is from: A = B (Hardcover)
This is a wonderful book, not only because it is well-written and useful, but because it is unlike any other math book I have ever encountered. It covers algorithms that automate the discovery and proof of combinatorial identities and formulas. It defines many important concepts, such as normal and canonical forms, and formulas in closed form--topics which are often thrown around casually but which many students never encounter precise definitions of. It is also a book about automated proof, and a philosophical text about the nature of proof and the discovery of proofs. In this respect, it is almost a must-read for any serious mathematician. It is available for free online, although it is certainly worth obtaining a hard-copy of, considering its usefulness and reasonable price.
As I said in my review of Generatingfunctionology, Wilf is an outstanding writer. This book tends to be a bit wordy and at times redundant, but the writing is amusing and often the redundancy serves to reinforce important and subtle points. This book is extremely easy to read; it requires very little prior background and would be accessible to many undergraduates. However, it covers material that a lot of people don't know, and it would probably be useful to most graduate students as well. The presentation of mathematical ideas is concise, and one will find it easy to skip around in the book as well.
The book has a rich interplay between algorithms and theory, and is cleverly designed so that the reader can focus on one while ignoring the other, but would benefit from studying both. The algorithms are astonishingly simple, especially given the amazing results they produce. But as I mentioned above, the true value of this book is on the "meta" level--this is a book that gets you to think about the nature of proof and the nature of mathematics. I think it would be a neat idea to create a course structured around this book.
I think this book is an excellent book for any mathematician to read--read it for yourself online; you may find yourself buying it!
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