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I have the following series: $$ f(t,q)=\sum_{k=0}^\infty (tq)^k (q;q)_k, $$ where $(q;q)_k$ is the $q$-pochhammer symbol, $0<q<1$. I suspect the series could be simplified, perhaps by identifying it with a special function of some kind, but I don't see how to do this, and would appreciate any help.

Thanks, -Alex

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Your series is equal to ${}_2\phi_{1}\big(\begin{smallmatrix}q,q \\0 \end{smallmatrix}\big|q;qt\big)$, where $$ {}_2\phi_{1}\big(\begin{smallmatrix}a,b \\c \end{smallmatrix}\big|q;z\big)=\sum_{n=0}^\infty \frac{(a;q)_n (b;q)_n}{(c;q)_n (q;q)_n} z^n $$ is the basic hypergeometric series. There is a vast amount of literature on these special functions; a detailed reference is Gasper & Rahman's Basic hypergeometric series. Maybe you will find some interesting property related to your work there.

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    $\begingroup$ Your post only introduces a notation for OP's functions, and then suggests that maybe an answer could be found somewhere in a book - but it's not guaranteed. This should have been a comment, at best. $\endgroup$
    – Alex M.
    Sep 2, 2017 at 13:48

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