Extension of a theorem of Duffin and Schaeffer

Abstract

Let $r_1,\dots ,r_s:{\mathbb{Z}}_{n⩾0}\to \mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\ddot{I}€\mathbb{Q}$ and let $F(z):={\sum }_{n⩾0}f(n)z^n$, where $f(n)\in \{r_1(n),\dots ,$ $r_s(n)\}$ for each $n⩾0$. 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.

2 pages