Given sequences $a_n$, $b_n$, $c_n=a_n b_n$, can you generally express a generating function of $c_n$ in terms of GF's of $a_n$ and $b_n$?

Question

Given the sequences $a_n$, $b_n$ and $c_n=a_n b_n$. Let $F(x) =\sum a_n x^n$, $G(x) =\sum b_n x^n$, $H(x) =\sum c_n x^n$ be the formal power series of the given sequences.

Can you generally express $H(x)$ as function of $F(x)$ and $G(x)$? What would that look like?