abbracciopj

2022-06-20

Consider the following function

$f(x)=\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{x}^{k}.$

Let us say I know ${a}_{k}$ belongs to a strict subset $F\subset \mathbb{R}$.

I want to characterize the set of sequences $\{{a}_{k}{\}}_{k=0}^{\mathrm{\infty}}$ for which $f(x)$ is a rational function in $x$.

Can someone point me to relevant results in literature?

$f(x)=\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{x}^{k}.$

Let us say I know ${a}_{k}$ belongs to a strict subset $F\subset \mathbb{R}$.

I want to characterize the set of sequences $\{{a}_{k}{\}}_{k=0}^{\mathrm{\infty}}$ for which $f(x)$ is a rational function in $x$.

Can someone point me to relevant results in literature?

jarakapak7

Beginner2022-06-21Added 14 answers

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.

$f(x)=\frac{p(x)}{q(x)},$

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.