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? I’ve been working on this infinite series from quite some time.

Any help would be appreciated. Thanks.