Generating series of rational$\times \exp($rational$)$

Question

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 $(a_n)_n$?

Answer 1

We have $$ \frac{d}{dx}\left ( f(x) e^{h(x)}\right) = \left( \frac{ f'(x)}{f(x)} + h'(x) \right) \left( f(x) e^{h(x)}\right).$$

Thus

$$ \sum_n a_n n x^{n-1} = \left( \frac{ f'(x)}{f(x)} + h'(x) \right) \sum_n a_n x^n .$$

If you express $\frac{ f'(x)}{f(x)} + h'(x)$ as a ratio of rational functions, put the denominator on the other side, and expand, you will get a linear recurrence relation whose coefficients are linear functions in $n$.

Conversely, any such linear recurrence relation can be put in this form, and therefore is associated to a rational function. The rational function is of the form $ \frac{ f'(x)}{f(x)} + h'(x)$ if and only if its residue at each point is an integer.