abbracciopj

2022-06-20

Consider the following function
$f\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_{k}{x}^{k}.$
Let us say I know ${a}_{k}$ belongs to a strict subset $F\subset \mathbb{R}$.
I want to characterize the set of sequences $\left\{{a}_{k}{\right\}}_{k=0}^{\mathrm{\infty }}$ for which $f\left(x\right)$ is a rational function in $x$.
Can someone point me to relevant results in literature?

jarakapak7

As
$f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},$
past a certain degree the coefficients of $f\left(x\right)q\left(x\right)$ vanish.
Hence except for the first few, the ${a}_{k}$ are constrained by a linear recurrence on $\text{deg}\left(q\right)+1$ terms.

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