Product of bivariate generating functions

Question

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?

Answer 1

$$c_{nm} = \sum_{i+j=n, k+l=m} a_{ik} b_{jl}.$$

This follows directly from basic properties of addition and multiplication and you should make sure to thoroughly understand this, for example by writing out the first few coefficients by hand. The generalization to any finite number of variables should be clear.