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.