Q-difference Polynomial - Generating Function

Generating Function

The generating function for these polynomials is of the type of generating function for Brenke polynomials, namely

where is the q-exponential:

$e_q(t)=\sum_{n=0}^\infty \frac{t^n}{_q!}= \sum_{n=0}^\infty \frac{t^n (1-q)^n}{(q;q)_n}.$

Here, is the q-factorial and

is the q-Pochhammer symbol. The function is arbitrary but assumed to have an expansion

Any such gives a sequence of q-difference polynomials.

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