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I've run into:

$$\sum_{x=1}^n {x^a\over 1-q^{x}}, \ s.t.\ q\in \mathbb N>1 \ or \ q\in (0, 1),\ a \in \mathbb N$$

I am interested mostly in the cases where $a = 1$ or $ a = 2$

Things I've done so far: reference a few places on basic hypergeometric series, not limited to something that looks remotely like what I'm interested in: Take a look at the "Simple Series" section, the first example

EDIT: I've made some progress, please see: JJacquelin's answer to this question, can someone help get their attention (I cannot comment yet). If we cannot, should not, or do not wish to get their attention through contact, perhaps explaining some of the manipulations in their answer might help us here. For example, pulling out the $1 \over m$ from the sum in the second to last line to obtain the integral shown, why was that done? What technique would apply if it were $1 \over m!$ instead?

Secondly, I've found that theta functions may be involved somehow:See GEdgar's answer here as well as Paramanand Singh's answer here.

user3108815
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    Do you really want $q \in \mathbb N$? If you're taking about series in powers of $q$, you should rather take $|q| < 1$. In any case, $q=1$ certainly doesn't work. – Robert Israel Jan 24 '19 at 05:13
  • @RobertIsrael Some thorough editing would have saved both of us some time with the way I asked. $0 < q<1$ does indeed work if I reframe the original line of thinking that led me here. $1$ should indeed be removed from the domain. – user3108815 Jan 24 '19 at 05:18
  • edited the question to reflect comments – user3108815 Jan 24 '19 at 08:11

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