Extension of a theorem of Duffin and Schaeffer

zbMATH Open1372.30002arXiv1706.02470MaRDI QIDQ5364154

Michael Coons

Publication date: 4 October 2017

Abstract: Let $r_1,ldots,r_s:mathbb{Z}_{ngeqslant0}omathbbC$ be linearly recurrent sequences whose associated eigenvalues have arguments in $pimathbbQ$ and let $F(z):=su{m}_{ngeqslant0}f(n)z^n$, where $$ $$ for each $ngeqslant0$. 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.

Full work available at URL: https://arxiv.org/abs/1706.02470

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zbMATH Keywords

power seriesrational functionDuffin-Schaeffer theorem

Mathematics Subject Classification ID

Recurrences (11B37) Transcendence theory of other special functions (11J91) Boundary behavior of power series in one complex variable; over-convergence (30B30)

Related Items (7)

An extension of the Derrida–Lebowitz–Speer–Spohn equation ⋮ The Duffin-Schaeffer conjecture with extra divergence. II ⋮ Title not available (Why is that?) ⋮ Title not available (Why is that?) ⋮ Tauberian theorems for sequences satisfying certain convolution relations ⋮ On the extension of a theorem of W. Dickmeis et al ⋮ Title not available (Why is that?)

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