Product of bivariate generating functions

The item of 2 univariate generating functions is merely offered by the Cauchy item.

$$ A(x) = \sum_{n=0} a_n x^n $$ $$ B(x) = \sum_{n=0} b_n x^n $$

$$ A(x)B(x) = C(x) = \sum_{n=0} x^n c_n $$ with $c_n = \sum_{k=0}^n a_k b_{n-k}$.

What is the resulting getting function for the bivariate instance?

$$ A(x,y) = \sum_{n=0}\sum_{m=0} a_{nm} x^n y^m $$ $$ B(x,y) = \sum_{n=0}\sum_{m=0} b_{nm} x^n y^m $$

$$ A(x,y)B(x,y) = C(x,y) = \sum_{n=0}\sum_{m=0} x^n y^m c_{nm} $$ What is $c_{nm}$, and also just how does this generalise to multivariate generating functions?

2019-12-02 03:12:36
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Answers: 1

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

This adheres to straight from standard buildings of enhancement and also reproduction and also you need to see to it to extensively recognize this, as an example by drawing up the first couple of coefficients by hand. The generalization to any kind of limited variety of variables need to be clear.

2019-12-03 05:08:36