is there a way to evaluate $\sum_{k=0}^{+\infty} \frac{x^k}{1-x^{2k+1}}$ in terms of popular functions or even in terms of the q-digamma function?
$0<x<1$
I tried to write the denominator as an geometric infinite series. Expression got cuter but didn't get easier. I can't see any integration or trick to work this one.
Wolfram gives me the partial sum in terms of the x-digamma function but it uses complex values in the digamma which makes hard for me to take the limit as I know nothing about this function.
So I gave it a better search in this forum and found that:
$\sum_{k=0}^{\infty} \frac{x^k}{1-x^{2k}} = L(x) - L(x^2)$
where $L(x) = \sum_{k=0}^{\infty} \frac{x^k}{1-x^{k}}$ is the Lambert series of $x$