TITLE: A not quite theta not quite basic hypergeometric function
QUESTION [8 upvotes]: The study of matrix quantum group coactions on the noncommutative disk algebra turns up the following series, which is a $q$-deformation of the negative binomial series, for integer $t\ge 0$, complex $z$ and $q\in[0,1]$:
$$\sum_{n\ge 0}(-1)^n q^{n(n-1)/2}\frac{[n+t]_q!}{[t]_q![n]_q!}\,z^n = 
\sum_{n\ge 0} (-1)^n q^{n(n-1)/2}\frac{(q^{1+t},q)_n}{(q,q)_n}\,z^n$$
written in terms of $q$-factorials or alternatively $q$-Pochhammer symbols. Now this is not a theta function (in the $t=0$ case), as the sum is over positive $n$ only. It is not a basic hypergeometric function, as the standard formula would not have the $q^{n(n-1)/2}$ factor. If anybody could give me a reference for such a function, or has other comments, I would be very grateful.

REPLY [4 votes]: Let's observe that $[n]_q!=1(1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})=\frac{(q)_n}{(1-q)^n}$ so that
\begin{align}
\sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{[n+t]_q!}{[n]_q![t]_q!}
&=\sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{(q)_{n+t}}{(q)_n(q)_t} \\
&=\sum_{n\geq0}(-1)^nq^{\binom{n}2}z^n\frac{(q^{t+1})_n}{(q)_n} \\
&=\lim_{\tau\rightarrow0}\,\,\,
{}_2 \phi_1 \left({{\frac1{\tau},q^{t+1}}\atop{\tau}}\middle|\,\,q,z\tau\right) \\
&=\lim_{\tau\rightarrow0}\,\,\frac{(q^{t+1})_{\infty}(z)_{\infty}}{(\tau)_{\infty}(z\tau)_{\infty}}\,\,\cdot\,\,
{}_2 \phi_1 \left({{\frac{\tau}{q^{t+1}},z\tau}\atop{z}}\middle|\,\,q,q^{t+1}\right) \\
&=(q^{t+1})_{\infty}(z)_{\infty}\,\sum_{n\geq0}\frac{q^{n(t+1)}}{(q)_n(z)_n};
\end{align}
where we have applied a Heine transformation in the penultimate step.
Perhaps other Heine transformations would lead to something more interesting.