Closed form for $\sum_{m \geq 1} (-1)^m q^{m(m+1)/2 + m \Delta}$? Is there a useful closed form for the following series ($|\Delta|$ is a small integer)?
$$f(q,\Delta) =\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 +  m \Delta}$$
It is a large-$n$ approximation of the polynomial $-[n+\Delta, n]_q$ discussed here.
EDIT: A more useful form, it turns out, is $ \tilde{f}(q,z) =\sum\limits_{m=1}^{\infty} (-1)^m q^{m(m+1)/2} z^m$. Its normal (non-$q$-analog) limit is trivial and appealing.

A fair bit of massaging is needed here.
$$\begin{align*}\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 +  m \Delta}&=\sum_{m=2}^{\infty} (-1)^{m-1} q^{m(m-1)/2}q^{(m-1)\Delta}\\&=-q^{-\Delta}\sum_{m=2}^{\infty} (-1)^m q^{m(m-1)/2}q^{m\Delta}\\&=q^{-\Delta}-1-q^{-\Delta}\sum_{m=0}^{\infty} (-1)^m q^{m(m-1)/2}q^{m\Delta}\\&=q^{-\Delta}-1-q^{-\Delta}\sum_{m=0}^{\infty}\frac{(q;q)_m}{(q;q)_m (0;q)_m} (-1)^m q^{m(m-1)/2}q^{m\Delta}\end{align*}$$
and finally we recognize the form of a basic hypergeometric function:
$$\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 +  m \Delta}=q^{-\Delta}-1-q^{-\Delta}{}_1 \phi_1\left({q \atop 0};q,q^\Delta\right)$$
Probably there is an easier expression in terms of Jacobi theta functions, but I haven't tried that route...
