Wellesley, MA: A. K. Peters. xii, 212 p. $ 39.00 (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. Wilf and Zeilberger 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. In this book, they detail where to find the packages that implement these algorithms in either Maple or Mathematica, they give examples of and instructions in how to use these packages, and they explain the motivation and theory behind the algorithms. The specific algorithms that are described are Sister Celine’s Method, an algorithm from the 1940’s that underlies most of the current research; Gosper’s Algorithm, the first of the powerful proof techniques to be implemented with a computer algebra package; Zeilberger’s Algorithm which extends and generalizes Gosper’s approach; the WZ Method which is guaranteed to provide a proof certificate for any correct identity for hypergeometric series and which can be used to determine whether or not a “closed form” exists for any given hypergeometric series. The book is also sprinkled with examples, exercises, and elaborations on the ideas that come into play.