# sequences and series – Closed form of \$sum_{z=1}^{infty}dfrac{z^n}{text{exp}(kz^m) – 1}\$

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.