Examples of generating functions
The following examples are in the spirit of George Pólya, who advocated learning mathematics by doing and re-capitulating as many examples and proofs as possible. The purpose of this article is to present common tricks of the trade in context, so that people may incorporate them into their knowledge.
Worked example A: basics
New generating functions can be created by extending simpler generating functions. For example, starting with
and replacing with , we obtain
Bivariate generating functions
One can define generating functions in several variables, for series with several indices. These are often called super generating functions, and for 2 variables are often called bivariate generating functions.
For instance, since is the generating function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients for all k and n. To do this, consider as itself a series (in n), and find the generating function in y that has these as coefficients. Since the generating function for is just , the generating function for the binomial coefficients is:
and the coefficient on is the binomial coefficient.
Worked example B: Fibonacci numbers
for this sequence. The generating function for the sequence (Fn−1) is xf and that of (Fn−2) is x2f. From the recurrence relation, we therefore see that the power series xf + x2f agrees with f except for the first two coefficients:
Taking these into account, we find that
(This is the crucial step; recurrence relations can almost always be translated into equations for the generating functions.) Solving this equation for f, we get
These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula