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.