What is the analytic expression for the infinite series f(x,y)?

[HolyPhia]

Homework Statement

I don't know if there is an analytic expression of this infinite series:
$$f(x,y)=\sum_{n=0}^{+\infty}\frac{x^n}{1-y^n}$$
here $$x,y<1$$

Homework Equations

This series is convergent, so maybe it can be expressed as some special function?

The Attempt at a Solution

I tried to differentiate $$f(x,y)$$ with respect to x and y, to find some relationship between $$\frac{\partial f(x,y)}{\partial x}$$and $$\frac{\partial f(x,y)}{\partial y}$$,etc. But it seems no help...

 

[lippyka]

And I also think the expression of this series may related to the zeta function, but I don't know how to connect them.Any help will be appreciated, thanks!