Definition:Generating Function

Let $$A = \left \langle {a_n}\right \rangle$$ be a sequence in $$\R$$.

Then $$G_A \left({z}\right) = \sum_{n \ge 0} a_n z^n$$ is called the generating function for the sequence $$A$$.

The mapping $$G_A \left({z}\right)$$ is defined for all $$z$$ for which the power series $$\sum_{n \ge 0} a_n z^n$$ is convergent.

The definition can be modified so that the lower limit of the summation is $$b$$ where $$b > 0$$ by assigning $$a_k = 0$$ where $$0 \le k < b$$.

Notation
When the sequence is understood, $$G \left({z}\right)$$ can be used.

Different authors use different symbols. $$\zeta \left({z}\right)$$ is sometimes seen but can be confused with the Riemann zeta function.

The variable is a dummy -- $$x$$ is often used instead.

Quote

 * A generating function is a clothesline on which we hang up a sequence of numbers for display.
 * Herbert Wilf, generatingfunctionology (1994)

Everybody else quotes it (it's the first line of the above book), so I don't see why this site should be any different.