Recently while working on some problems, I came across this beautiful infinite series:
$$\sum_{z=1}^{\infty}\dfrac{z^n}{\text{exp}(kz^m) - 1}$$
where $m,n \in \mathbb{N} $. Does there exist a closed form for this infinite series? What happens if $m = n$? Can we find a closed form when $m = 2, n = 4?$ or when $m = 5, n=6$? What would be a general approach? Any idea? Can we use complex analytic methods such as residue theorem and stuff to evaluate this series? I've been working on this infinite series from quite some time.
Any help would be appreciated. Thanks in advance.