General introduction to generating functions

This book makes extensive use of generating functions. In that respect the discussions here are consistent with the approach that condensed matter physicists generally take when calculating properties of the random walk as it relates to problems of contemporary interest. This chapter is devoted to a discussion of the generating function and to an exploration of some of the ways in which the generating function method can be put to use in the study of the random walk. Many of the arguments in later chapters will call upon techniques and results that will be developed in the pages to follow. Thus, the reader is strongly urged to pay close attention to the discussion that follows, as topics and techniques that are introduced here will crop up repeatedly later on.

What is a generating function?

The generating function is a mathematical stratagem that simplifies a number of problems. Its range of applicability extends far beyond the mathematics of the random walk. Readers who have had an introduction to ordinary differential equations will have already seen examples of the use of the method of the generating function in the study of special functions. The generating function also plays a central role in graph theory and in the study of combinatorics, percolation theory, classical and quantum field theory and a myriad of other applications in physics and mathematics. Briefly, a generating function is a mathematical expression, depending on one or more variables, that admits a power series expansion. The coefficients of the expansion are the members of a family, or sequence, of numbers or functions.