Product of Generating Functions/General Rule

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Theorem

Let $\map {G_0} z, \map {G_1} z, \map {G_2} z, \ldots$ be any number of generating functions (up to countably infinite) for the sequences $\sequence {a_0 n}, \sequence {a_1 n}, \sequence {a_2 n}, \ldots$

Then:

\(\ds \prod_{j \mathop \ge 0} \map {G_j} z\) \(=\) \(\ds \prod_{j \mathop \ge 0} \sum_{k \mathop \ge 0} a_{j k} z^k\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} z^n \sum_{\substack {k_0, k_1, k_2, \ldots \mathop \ge 0 \\ k_0 \mathop + k_1 \mathop + \mathop \cdots \mathop = n} } \paren {\prod_{j \mathop \ge 0} a_{j k} }\)


Proof


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Sources

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