# 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

, Volume 186 (2) – May 24, 2017
13 pages

/lp/springer_journal/rational-transcendental-dichotomy-of-power-series-with-a-restriction-8uuL0K9CdY
Publisher
Springer Journals
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

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