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Lectures on Generating Functions

American Mathematical Society
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Lando's Lectures on Generating Functions is distinguished by two closely intertwined features: It is driven by very, very interesting problems and examples; and it takes off in directions which are perhaps a little unusual in an introductory text on generating functions, fitted in the larger framework of combinatorics. The former feature is already abundantly evident from the first sentence of the book's preface: "After multiplying by (2n-1)! the coefficient of [the n-th term] in the power series expansion of the function tan(x) becomes a positive integer..." Thus, without naming names — and thereby giving away the tantalizing secret — Lando begins the book, even before the first chapter, with a veiled allusion to a candidate for the most famously named numbers in number theory. And he can hardly contain his enthusiasm (which is indeed quickly infectious) as he goes on to the second paragraph of the preface: "Mathematicians of the 18th and 19th centuries knew functions 'personally'..." Surely this is an irresistible challenge to join him in the recovery of a lost art!

Regarding the second property, the author's particular choice of topics, it's really all an immense amount of fun. After dealing very effectively with the requisite background concerning fundamental properties of elementary generating functions, Lando hits, for example, the Fibonacci numbers, the Catalan numbers, and formal grammars. Then, quickly introducing more and more (beautiful) machinery as he goes along, he gets to Pascal's triangle, the Dyck triangle, the Bernoulli-Euler triangle, and the Euler numbers, all in the context of generating functions of several variables. Very good stuff! And the last chapters of the book deal with such gems as the theory of partitions, continued fractions, and enumeration problems for embedded graphs.

The book is very well suited to self study or use in a seminar for a hand-picked audience. It is beautifully written, even if the exposition is a bit on the terse side, and it is certainly indicated that the reader or student should possess a fair amount of "mathematical maturity." This having been said, reading Lando's book is a trip well worth taking for any one interested in the topic of generating functions and the arithmetically well-endowed numbers or related objects, such as certain ordinary differential equations (see paragraph 5.8) which they describe.

Michael Berg ( is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Date Received: 
Sunday, May 1, 2005
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S. K. Lando
Student Mathematical Library 23
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Michael Berg
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* Formal power series and generating functions. Operations with formal power series. Elementary generating functions
* Generating functions for well-known sequences
* Unambiguous formal grammars. The Lagrange theorem
* Analytic properties of functions represented as power series and their asymptotics of their coefficients
* Generating functions of several variables
* Partitions and decompositions
* Dirichlet generating functions and the inclusion-exclusion principle
* Enumeration of embedded graphs
* Final and bibliographical remarks
* Bibliography
* Index
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Monday, January 17, 2011