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