Examples of generating functions

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.

Worked example A: basics

New generating functions can be created by extending simpler generating functions. For example, starting with

G(1;x)=\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

and replacing $x$ with $ax$ , we obtain

G(1;ax)=\frac{1}{1-ax} = 1+(ax)+(ax)^2+\cdots+(ax)^n+\cdots =\sum_{n=0}^{\infty} a^n x^n = G(a^n;x).

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 $(1+x)^n$ is the generating function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients $\binom{n}{k}$ for all k and n. To do this, consider $(1+x)^n$ as itself a series (in n), and find the generating function in y that has these as coefficients. Since the generating function for $a^n$ is just $1/(1-ay)$ , the generating function for the binomial coefficients is:

\frac{1}{1-(1+x)y}=1+(1+x)y+(1+x)^2y^2+\dots,

and the coefficient on $x^ky^n$ is the $\binom{n}{k}$ binomial coefficient.

Worked example B: Fibonacci numbers

Consider the problem of finding a closed formula for the Fibonacci numbers F_n defined by F₀ = 0, F₁ = 1, and F_n = F_n−1 + F_n−2 for n ≥ 2. We form the ordinary generating function

f = \sum_{n \ge 0} F_n x^n

for this sequence. The generating function for the sequence (F_n−1) is xf and that of (F_n−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:

\begin{array}{rcrcrcrcrcrcr} f & = & F_0x^0 & + & F_1x^1 & + & F_2x^2 & + & \cdots & + & F_ix^i & + &\cdots\\ xf & = & & & F_0x^1 & + & F_1x^2 & + & \cdots & + &F_{i-1}x^i & + &\cdots\\ x^2f & = & & & & & F_0x^2 & + & \cdots & + &F_{i-2}x^i & +&\cdots\\ (x+x^2)f & = & & & F_0x^1 & + & (F_0+F_1)x^2 & + & \cdots & + & (F_{i-1}+F_{i-2})x^i & +&\cdots\\ & = & & & & & F_2x^2 & + & \cdots & + & F_ix^i & +& \cdots\\ \end{array}

Taking these into account, we find that

f = xf + x^2 f + x . \,\!

(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

f = \frac{x} {1 - x - x^2} .

The denominator can be factored using the golden ratio φ₁ = (1 + √5)/2 and φ₂ = (1 − √5)/2, and the technique of partial fraction decomposition yields

f = \frac{1}{\sqrt{5}} \left (\frac{1}{1-\varphi_1 x} - \frac{1} {1- \varphi_2 x} \right ) .

These two formal power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula

F_n = \frac{1} {\sqrt{5}} (\varphi_1^n - \varphi_2^n).

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