I've run into:

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

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

I'm looking for something short and simple, ideally. Of course, if that cannot be done, we'll settle for computationally efficient with a bounded number of terms for any input.

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.

I've run into:

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

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

I'm looking for formulae that are "short and simple", ideally. Of course, if that cannot be done, I'll settle for computationally efficient with a static number of terms regardless of input.

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.

I've run into:

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

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.

I've run into:

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

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

I'm looking for something short and simple, ideally. Of course, if that cannot be done, we'll settle for computationally efficient with a bounded number of terms for any input.

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.