\begin{definition}[Definition:Generating Function/Doubly Subscripted Sequence]
Let $A = \sequence {a_{m, n} }$ be a doubly subscripted sequence in $\R$ for $m, n \in \Z_{\ge 0}$.
Then $\ds \map {G_A} {w, z} = \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ is called the '''generating function''' for the sequence $A$.
The mapping $\map {G_A} {w, z}$ is defined for all $w$ and $z$ for which the power series $\ds \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ 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 $0 \le k < b$.
\end{definition}