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Extension of a Theorem of Duffin and Schaeffer
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Michael Coons

School of Mathematical and Physical Sciences

The University of Newcastle

Callaghan, NSW 2308

Australia

**Abstract:**

Let *r*_{1},..., *r*_{s}:
**Z**_{n≥0} → **C** be linearly
recurrent sequences whose associated eigenvalues have arguments in
π**Q**
and let *F*(*z*) := Σ_{n ≥ 0} *f*(*n*)*z*^{n}, where
*f*(*n*) ∈ {*r*_{1}(*n*),..., *r*_{s}(*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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