Say I'm rolling an $m$-sided die and an $n$-sided die once.
The probability distribution of all possible sums of the two dice rolls can be determined by normalizing the coefficients the product of $f_m(x) = x + \dots + x^m$ and $f_n(x) = x + \dots + x^n$. Is there an analogous way to determine the probability distribution of the products of the two dice rolls via operations on polynomials? Is there maybe even a more general way to do this for arbitrary functions? Ideally something that can actually be computed without having to solve an integral or similar.