The product of two univariate generating functions is simply given by the Cauchy product.
$ A(x) = \sum_{n=0} a_n x^n $ $ B(x) = \sum_{n=0} b_n x^n $
$ A(x)B(x) = C(x) = \sum_{n=0} x^n c_n $ with $c_n = \sum_{k=0}^n a_k b_{n-k}$.
What is the resulting generating function for the bivariate case?
$ A(x,y) = \sum_{n=0}\sum_{m=0} a_{nm} x^n y^m $ $ B(x,y) = \sum_{n=0}\sum_{m=0} b_{nm} x^n y^m $
$ A(x,y)B(x,y) = C(x,y) = \sum_{n=0}\sum_{m=0} x^n y^m c_{nm} $ What is $c_{nm}$, and how does this generalize to multivariate generating functions?