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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.

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  • $\begingroup$ Any context for where this q-series came from? $\endgroup$ – Somos Sep 25 '18 at 13:17
  • $\begingroup$ 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. $\endgroup$ – Aran Sep 25 '18 at 13:21
  • $\begingroup$ 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. $\endgroup$ – Hazem Orabi Sep 25 '18 at 18:49
  • $\begingroup$ @Hazem Orabi Apologies, I mistook voting up for accepting answers! $\endgroup$ – Aran Sep 25 '18 at 19:32

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