Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.10

The Rational-Transcendental Dichotomy of Mahler Functions

Jason P. Bell
Department of Mathematics
Simon Fraser University
Burnaby, BC

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

Eric Rowland
Université du Québec à Montréal
Montréal, QC


In this paper, we give a new proof of a result due to Bézivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our method of proof, we also provide a new proof of a Pólya-Carlson type result for Mahler functions due to Randé; that is, a Mahler function which is meromorphic in the unit disk is either rational or has the unit circle as a natural boundary.

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(Concerned with sequence A010060.)

Received July 3 2012; revised version received October 2 2012. Published in Journal of Integer Sequences, March 2 2013.

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