Polynomial identity for a sum

Question

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial

$$h(x) = \sum_{k=0}^{2n}C_k x^k \quad \text{ where } \quad C_k = \sum_{\substack{0\leq i, j\leq n \\ i + j = k}}A_i B_j$$

can be expressed as $$h(x) = f(x)\cdot g(x).$$

Is there a similar method to express the polynomial

$$r(x) = \sum_{k=0}^{n^2}D_k x^k \quad \text{ where } \quad D_k = \sum_{\substack{0\leq i, j\leq n \\ i\cdot j = k}}A_i B_j$$

like $r(x) = \mathcal{F}\thinspace (f(x), g(x))$ ?

Answer 1

I think this would be along the lines of $F(f(x),g(x))=\sum_{j=0}^nf (x)^jg (x)^{n-j}$

Answer 2

There is a partial answer. Given the power series expansion for $f(x)$ in the question, the function $r(x)$ can be expressed as follows.

$$r(x) = \sum_{i=0}^n A_i g(x^i)$$

I am unable to make the form more compact that this.