Journal of Integer Sequences, Vol. 20 (2017), Article 17.9.4

Extension of a Theorem of Duffin and Schaeffer

Michael Coons
School of Mathematical and Physical Sciences
The University of Newcastle
Callaghan, NSW 2308


Let r1,..., rs: Zn≥0C be linearly recurrent sequences whose associated eigenvalues have arguments in πQ and let F(z) := Σn ≥ 0 f(n)zn, where f(n) ∈ {r1(n),..., rs(n)} for each n ≥ 0. We prove that if F(z) is bounded in a sector of its disk of convergence, then 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.

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Received June 8 2017; revised versions received August 9 2017; September 5 2017; September 11 2017. Published in Journal of Integer Sequences, September 15 2017.

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