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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.

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    I'm rather unskilled in manipulating modular forms (sorry), but you do have a good reason to suspect that Ramanujan has considered series of this sort; it does look like something he'd have looked at...2011-10-19

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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...

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    I thnik I've got an idea - take $q=e^{-\epsilon}$ and convert the sum int he limit of $\epsilon \to 0 $ to an integral. I looks like an error integral, should be expandable then in $\epsilon$ to the leading order.2011-10-18