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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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up vote 2 down vote accepted

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

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.

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The generalization to infinitely many variables, on the other hand... –  Qiaochu Yuan Dec 8 '10 at 0:13
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