Which Pólya's theorem does this refer to? Searching, I find the Pólya enumeration theorem.

@bubba That is true, however the involved expressions become rather involved very quickly, and I don't know of a way to do it generically for all values of $p$.

@WillyGoat No, I'm still wondering.

There are many ways of doing bivariate approximation. Do you have any indication why $G(m(x)+n(y))$ might be a good representation for your functions? In any case, I think you might find more help on scicomp.stackexchange.com (at least if you give more detail on your problem).

There's a small typo here: the $n$'s should be either all capital or not as they all refer to the polynomial degree.