Doubly Sequenced Generating Function for Binomial Coefficients

Theorem

Let $\sequence {a_{m n} }$ be the doubly subscripted sequence defined as:

$\forall m, n \in \N_{\ge 0}: a_{m n} = \dbinom n m$

where $\dbinom n m$ denotes a binomial coefficient.

Then the generating function for $\sequence {a_{m n} }$ is given as:

$\map G {w, z} = \dfrac 1 {1 - z - w z}$

$\ds \map G {w, z}$

$=$

$\ds \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$

$\ds $

$=$

$\ds \sum_{m, \, n \mathop \ge 0} \dbinom n m w^m z^n$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \paren {1 + w}^n z^n$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \paren {\paren {1 + w} z}^n$

$\ds $

$=$

$\ds \dfrac 1 {1 - \paren {1 + w} z}$

$\ds $

$=$

$\ds \dfrac 1 {1 - z - w z}$

$\blacksquare$