Product of bivariate generating functions

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?

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