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.