This book grabs the reader from its first sentence: “A generating function is a clothesline on which we hang up a sequence of numbers for display.” In five well-written chapters, the author exposes the interplay between the discrete and the continuous, taking the uninitiated from the basics to significant applications of generating function techniques. He discusses power series, exponential series, and Dirichlet series. After a discussion of formal power series, he discusses the analytic theory resulting from replacing the indeterminate x by a complex number z, providing a neat discussion of convergence in the complex plane as background. Some parts of the book require knowledge of complex integration theory, but this use is minimal. The exercises are excellent and frequently introduce extensions or applications of the theory discussed previously. The inclusion of all answers/solutions at the back of the book enhances the usefulness of this volume for self-study.
Along the way, Wilf’s book applies generatingfunctionology to problems in probability/statistics, algebra/number theory, recurrences, and combinatorics in general.
Wilf has an enlightening treatment of what he calls the “Snake Oil Method for Doing Combinatorial Sums” — a five-step program he illustrates with seven marvelous examples. He follows this with a wonderful introduction to Wilf-Zeilberger (WZ) pairs, a relatively recent technique for proving combinatorial identities. This method lends itself to implementation on a CAS. The last chapter provides useful tools, especially the Lagrange Inversion Formula, for determining the asymptotic growth rate of a sequence. There is an appendix on using Mathematica and Maple to manipulate formal power series, solve recurrences, and find asymptotic expansions of functions.
I must point out that the differences between this edition and the second edition, which was available from another publisher and could also be downloaded from the author’s home page, are minimal. There are two new sections (comprising roughly four pages) on the probability of the occurrence of a string in a series of coin tosses and the number of strings of n bits, m of which are 1’s, that contain no block of k consecutive 1’s. Also, there has been a minor updating of the references from the second edition. Essentially, it’s the same book, but in hard copy — and, at the time of this review, the second edition is still available for downloading!
Henry Ricardo (email@example.com) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.