The following examples of
generating functions 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 ways of creating 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
Consider the problem of finding a
closed formula for the
Fibonacci numbersFn defined by F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2. We form the ordinary generating function
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
Worked example C: Number of ways to make change
The number of unordered ways an to make change for n cents using coins with values 1, 5, 10, and 25 is given by the generating function
For example there are two unordered ways to make change for 6 cents; one way is six 1-cent coins, the other way is one 1-cent coin and one 5-cent coin. See OEIS:
On the other hand, the number of ordered ways bn to make change for n cents using coins with values 1, 5, 10, and 25 is given by the generating function
For example there are three ordered ways to make change for 6 cents; one way is six 1-cent coins, a second way is one 1-cent coin and one 5-cent coin, and a third way is one 5-cent coin and one 1-cent coin. Compare to OEIS:
A114044, which differs from this example by also including coins with values 50 and 100.