Rational-transcendental dichotomy of power series with a restriction on coefficients

Rational-transcendental dichotomy of power series with a restriction on coefficients In 1906, Fatou proved that a rational-transcendental dichotomy holds for power series whose coefficients are taken from a finite set of complex numbers. In 1945, Duffin and Schaeffer proved that if a power series with coefficients from a finite subset of $${\mathbb {C}}$$ C is bounded in a sector of the unit circle, then it must be a rational function. Duffin and Schaeffer’s result, from which Fatou’s theorem can be derived, is a generalization of a result of Szegö. In this paper, we investigate power series with coefficients uniformly taken from finitely many polynomial sequences, that is, a power series whose every n-th term coefficient is taken from a set $$\{P_1(n),\ldots ,P_r(n)\}$$ { P 1 ( n ) , … , P r ( n ) } for some given polynomials $$P_1(z),\ldots ,P_r(z)$$ P 1 ( z ) , … , P r ( z ) .We prove that if a power series of this form over any field of characteristic zero is D-finite, then it is a rational power series. As a byproduct, we obtain that the rational-transcendental dichotomy holds for power series of this form, which is more general than Fatou’s result. We also show a generalization of Duffin and Schaeffer’s result that states as follows: If a power series in $${\mathbb {C}}[[z]]$$ C [ [ z ] ] with coefficients uniformly taken from finitely many polynomial sequences is bounded in a sector of the unit circle, then it is already rational. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

Rational-transcendental dichotomy of power series with a restriction on coefficients

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Publisher
Springer Vienna
Copyright
Copyright © 2017 by Springer-Verlag Wien
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1064-8
Publisher site
See Article on Publisher Site

Abstract

In 1906, Fatou proved that a rational-transcendental dichotomy holds for power series whose coefficients are taken from a finite set of complex numbers. In 1945, Duffin and Schaeffer proved that if a power series with coefficients from a finite subset of $${\mathbb {C}}$$ C is bounded in a sector of the unit circle, then it must be a rational function. Duffin and Schaeffer’s result, from which Fatou’s theorem can be derived, is a generalization of a result of Szegö. In this paper, we investigate power series with coefficients uniformly taken from finitely many polynomial sequences, that is, a power series whose every n-th term coefficient is taken from a set $$\{P_1(n),\ldots ,P_r(n)\}$$ { P 1 ( n ) , … , P r ( n ) } for some given polynomials $$P_1(z),\ldots ,P_r(z)$$ P 1 ( z ) , … , P r ( z ) .We prove that if a power series of this form over any field of characteristic zero is D-finite, then it is a rational power series. As a byproduct, we obtain that the rational-transcendental dichotomy holds for power series of this form, which is more general than Fatou’s result. We also show a generalization of Duffin and Schaeffer’s result that states as follows: If a power series in $${\mathbb {C}}[[z]]$$ C [ [ z ] ] with coefficients uniformly taken from finitely many polynomial sequences is bounded in a sector of the unit circle, then it is already rational.

Journal

Monatshefte f�r MathematikSpringer Journals

Published: May 24, 2017

References

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