# Extension of a theorem of Duffin and Schaeffer

## TL;DR

This paper extends a recent theorem by showing that certain bounded generating functions, constructed from linearly recurrent sequences with rational argument eigenvalues, are rational functions, generalizing previous polynomial-based results.

## Contribution

It generalizes Tang and Wang's result by proving that bounded generating functions from specific linearly recurrent sequences are rational, broadening the class of sequences considered.

## Key findings

- Bounded generating functions are rational functions.
- Extension from polynomial-based sequences to linearly recurrent sequences.
- Applicable to sequences with eigenvalues having rational multiples of pi arguments.

## Abstract

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$ $r_s(n)\}$ for each $n\geqslant 0$. We prove that if $F(z)$ is bounded in a sector of its disk of convergence, it is a rational function. This extends a very recent result of Tang and Wang, who gave the analogous result when the sequence $f(n)$ takes on values of finitely many polynomials.