Definition:Generating Function/Doubly Subscripted Sequence

Definition
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 $0 \le k < b$.