Generating functions

Abstract

You may recall from a calculus class the geometric series:

$$\begin{aligned} \frac{1}{1-z} = 1+z + z^2 + z^3 + \cdots + z^k + \cdots = \sum _{k\ge 0} z^k. \end{aligned}$$

The coefficients in the (MacLaurin) series expansion of a function F(z) define a sequence of numbers. Generally, if F(z) has MacLaurin series

$$\begin{aligned} \sum _{k\ge 0} a_k z^k = a_0 + a_1 z + a_2z^2 + \cdots + a_k z^k + \cdots , \end{aligned}$$

we can relate the function F with the sequence $a_0, a_1, a_2,\ldots$. So the function $1/(1-z)$ encodes the rather boring sequence $1, 1, 1,\ldots$.

“A generating function is a clothesline on which we hang up a sequence of numbers for display.”

–Herb Wilf