Doubly Sequenced Generating Function for Binomial Coefficients

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Theorem

Let $\left \langle {a_{m n} }\right \rangle$ 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 $\left \langle {a_{m n} }\right \rangle$ is given as:

$G \left({w, z}\right) = \dfrac 1 {1 - z - w z}$


Proof

\(\displaystyle G \left({w, z}\right)\) \(=\) \(\displaystyle \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n\) Definition of Generating Function for Doubly Subscripted Sequence
\(\displaystyle \) \(=\) \(\displaystyle \sum_{m, \, n \mathop \ge 0} \dbinom n m w^m z^n\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \ge 0} \left({1 + w}\right)^n z^n\) Binomial Theorem
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \ge 0} \left({\left({1 + w}\right) z}\right)^n\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {1 - \left({1 + w}\right) z}\) Sum of Infinite Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {1 - z - wz}\)

$\blacksquare$


Sources