Sum with reciprocal general term.

Question

Given a function $f$ expressed as follows as a formal power series (with $a_k\neq 0$) $$ f(x)=\sum_{k\geq0}a_kx^k, $$ I was wondering if there is some way to relate the latter with the function $g$ $$ g(x)=\sum_{k\geq0}\frac{x^k}{a_k}. $$

Answer 1

The general situation is that the only relation between the two is $$f^{(n)}(0)g^{(n)}(0)=(n!)^2$$

Answer 2

You're asking for a general description of the inverse under the Hadamard product of a known function, presumably in terms of standard operations like derivatives and reciprocals. There seems to be no hope for this. If it existed, surely the standard introductions would mention it (but of course it's very difficult to prove something like this doesn't exist).

Attempting to describe the Hadamard inverse of a rational function might be interesting.