Consider the following function $$f(x) = \sum_{k=0}^{\infty}a_kx^k.$$
Let us say I know $a_k$ belongs to a strict subset $F\subset\mathbb{R}$.
I want to characterize the set of sequences $\{a_k\}_{k=0}^\infty$ for which $f(x)$ is a rational function in $x$.
Can someone point me to relevant results in literature?
As
$$f(x)=\frac{p(x)}{q(x)},$$ past a certain degree the coefficients of $f(x)q(x)$ vanish.
Hence except for the first few, the $a_k$ are constrained by a linear recurrence on $\text{deg}(q)+1$ terms.