Is there a name for the following type of infinite series function?

Question

Consider the function:

$$f(x,y,z) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^{x}}{n^{y}(n+m)^{z}} \quad .$$

Questions:

1. Does this (family of) function(s) have a name? Is there any literature on it?
2. What about the special case in which $y = z$ ?
3. What about the particular value in which $x=2$ and $y = z = 4$ ?

Answer 1

If the Witten Zeta-function is defined by

$$W(r,s,t)=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{ 1 }{n^{r} m^{s}(n+m)^{t}}$$

then your function $f(x,y,z)$ is equivalent to $W(r,-s,t)$ (assuming $x$ and $y$ are swapped in your function above)

See Thirty-two Goldbach Variations by Jonathan M. Borwein, David M. Bradley

Maybe it will be useful to search the literature on MORDELL-TORNHEIM SUMS and MULTIPLE ZETA VALUES

If the double sum is absolutely convergent you can reverse the order of summation if that helps find a solution.