It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant coefficients of length equal to the degree of the denumerator, and conversely.

My question regards similar properties of generating series of $f(x)\exp(h(x))=\sum_{n\geq0} a_nx^n$ with $f, h\in \mathbb C(x)$ without pole at $0$. Is there any nice feature one should be aware about finite determinacy of the sequence $(a_n)_n$?