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Transcendence tests for Mahler functions

Authors: Jason P. Bell and Michael Coons
Journal: Proc. Amer. Math. Soc. 145 (2017), 1061-1070
MSC (2010): Primary 11J91; Secondary 39A06, 30B30
Published electronically: September 15, 2016
MathSciNet review: 3589306
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Abstract: We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue $ \lambda _F$ of a Mahler function $ F(z)$ and develop a quick test for the transcendence of $ F(z)$ over $ \mathbb{C}(z)$, which is determined by the value of the eigenvalue $ \lambda _F$. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of $ F(z)$. We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.

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Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Michael Coons
Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales 2308, Australia

Keywords: Transcendence, Mahler functions, radial asymptotics
Received by editor(s): November 1, 2015
Received by editor(s) in revised form: May 17, 2016
Published electronically: September 15, 2016
Additional Notes: The research of the first author was supported by NSERC grant 31-611456
The research of the second author was supported by ARC grant DE140100223.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society