Generating functionology, third edition

There are not too many books devoted entirely to generating functions (GF). Even less of them have appeared in this century. One of them is this book. It is the third edition of a very popular book based on the author’s lectures at the University of Pennsylvania. GF are an indispensable tool for discrete mathematics. Roots of the use of GF are deep. Let us recall that the Binet formula for Fibonacci numbers was discovered using a generating function method by Moivre in 1718. At the end of the eighteenth century GF were installed as a basic method of probabilistic computations by Laplace.

The book gently introduces the reader to the way that GF are used for solving problems. It also underlines another role of GF: they form a natural bridge between two seemingly distant areas of continuous and discrete mathematics. Despite the fact that the author says that he tried only to communicate some of the main ideas on the subject, the book gives the beginner a surprisingly broad view of many different uses of GF. The author presents applications of GF ranging from set partitions, the money changing problem and graph theory to relations of GF to unimodality, convexity and proofs of some congruences. For about the first seventy pages basic notions and notations are introduced and then various problems are solved. Each of the five chapters contains exercises (all solutions are provided at the end of the book). The book is very readable and it should not be missing from any university library. In particular I would like to quote that the author nicely compares a GF to a “clothesline on which we hang up a sequence of numbers to display”.

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