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I am interested in the sum of a Hadamard product of generating functions.

If we are given $n$ functions, where $0 < i \leq n$, of the form:

$f_i(x_i) = \sum_{j=0}^m{c_{i,j} x_i^j}.$

The hadamard product is defined as:

$g(x) = \sum_{j=0}^m{( \prod_{i=0}^n {c_{i,j}})x^j}.$

It's essentially the same as going through all the generating functions simultaneously and multiplying all the $j$th coefficients together to obtain a new coefficient.

Some given information

I have polynomials; I think of them as finite length generating functions. I know that all of the coefficients are natural numbers. I'd like the sum of the resulting coefficients.

There's a trick. Calculating the Hadamard product generates an extremely complicated expression. I would like to avoid dealing with this expression explicitly, if possible.

I want to know ways that I could do this.

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    Don't know if this will help but, the Hadamard product of two rational functions is rational (Theorem 2.4 of "Lectures on Generating Functions" by S. K. Lando, AMS publication stml-23).2011-09-20

0 Answers 0