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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Hypertranscendence and linear difference equations

Authors: Boris Adamczewski, Thomas Dreyfus and Charlotte Hardouin
Journal: J. Amer. Math. Soc.
MSC (2020): Primary 39A06, 12H05
Published electronically: January 20, 2021
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Abstract: After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator $ x\mapsto x+h$ ( $ h\in \mathbb{C}^*$), the $ q$-difference operator $ x\mapsto qx$ ( $ q\in \mathbb{C}^*$ not a root of unity), and the Mahler operator $ x\mapsto x^p$ ($ p\geq 2$ integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable $ x$ with complex coefficients (or in the variable $ 1/x$ in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.

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Additional Information

Boris Adamczewski
Affiliation: Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France

Thomas Dreyfus
Affiliation: Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg, France

Charlotte Hardouin
Affiliation: Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, UPS, F-31062 Toulouse Cedex 9, France

Received by editor(s): October 7, 2019
Received by editor(s) in revised form: June 22, 2020
Published electronically: January 20, 2021
Additional Notes: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under the Grant Agreement No 648132, and from the ANR De rerum natura project (ANR-19-CE40-0018).
Article copyright: © Copyright 2021 American Mathematical Society