Product of Generating Functions/General Rule

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

This theorem requires a proof.
In particular: There may be some general convolution result that can be used. Good luck Jim.
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Sources

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(9)$