Definition:Generating Function/Doubly Subscripted Sequence

Definition
Let $A = \left \langle {a_{m, n} }\right \rangle$ be a doubly subscripted sequence in $\R$ for $m, n \in \Z_{\ge 0}$.

Then $\displaystyle G_A \left({w, z}\right) = \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ is called the generating function for the sequence $A$.

The mapping $G_A \left({z}\right)$ is defined for all $w$ and $z$ for which the power series $\displaystyle \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ 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 $0 \le k < b$.