# A not quite theta not quite basic hypergeometric function

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.

"Basic hypergeometric series are series $\sum c_n$, with $c_{n+1}/c_n$ a rational function of $q^n$, for a fixed parameter $q$, which is usually taken to satisfy $|q|<1$, but at other times is a power of a prime."

This quote is from the Forward (by Richard Askey) to "Basic Hypergeometric Series", by Gaspar and Rahman, p. xvi, which can be found on the preview here: https://www.amazon.com/Basic-Hypergeometric-Encyclopedia-Mathematics-Applications/dp/0521833574/ref=sr_1_1?ie=UTF8&qid=1495081420&sr=8-1&keywords=basic+hypergeometric+series

Your series above, as well as theta series, are both basic hypergeometric series.

There are different conventions for their notation; some such as Gasper and Rahman's notation make (powers of) the factor $(-1)^n q^{(n-1)n/2}$ explicit for some cases, but not others. In any event, regardless of the notation, these factors arise by changing variables and considering special cases. For example, the series in your question arises from Heine's series $$_2\phi_1(a,b,c;q;z) = \sum_{n\geq 0} \frac{(a)_n(b)_n}{(c)_n(q)_n}z^n$$ by setting $a=q^{1+t}$, $c=0$, making the change of variables $b\mapsto 1/b$, then $z\mapsto bz$, and finally setting $b=0$.

The book linked above is essentially the bible for these series.

Had you looked at the modern definition of a $q$-hypergeometric function (from Gasper/Rahman, as noted in the previous answer), you would see that if the numerator and denominator parameters are equal, then there certainly is a $(-1)^n q^\binom{n}{2}$ factor in the sum's general term. Thus, with that definition, your function can be expressed as

$${}_1 \phi_1 \left({{q^{t+1}}\atop{0}}\middle|q;z\right)$$

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.