# Evaluation of complicated infinite q-series

I have the following q-series that I would like to explicitly evaluate, which is dependent on two numbers, $$n$$ and $$m$$, where $$m$$ is odd:

$$\sum_{a=-\infty}^{+\infty} \frac{q^{2a}}{(1-q^{2a+m})(1-q^{2a-m-2})}\frac{1+2n-2a}{(1-q^{2(1+2n-2a)})}$$

$$|q|<1$$ and application of the ratio test in both the positive and negative directions imply that the infinite sum converges, as in the positive direction we find the ratio tends to $$\|q\|^6<1$$ and in the negative direction the ratio tends to $$\|q\|^2<1$$.

The most natural step for me would be to split the factors into the denominator up. This gives 3 partial fractions (with a factor independent of the summand), which go like

$$\frac{1}{q^{-2a}-q^{b}}$$ where $$b$$ is odd, and one partial fraction where there is a plus in the denominator.

Unfortunately, trying to then compute the resulting sums doesn't seem to be possible because is series diverges.

Any help with this problem would be greatly appreciated.

• Any context for where this q-series came from? – Somos Sep 25 '18 at 13:17
• I'm trying to evaluate and simplify the integral given in math.stackexchange.com/questions/2891122/…. Specifically I have evaluated the summation of the $a_i$ and $b_i$ indices and am now trying to evaluate the summation over the last set of indices, from which this sum turned up. – Aran Sep 25 '18 at 13:21
• Hi friend, I notice that you haven't accepted any answer for all your questions! Accepting correct answers shall encourage others to help you. Thanks. – Hazem Orabi Sep 25 '18 at 18:49
• @Hazem Orabi Apologies, I mistook voting up for accepting answers! – Aran Sep 25 '18 at 19:32