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Values of globally bounded $ G$-functions


Authors: S. Fischler and T. Rivoal
Journal: Proc. Amer. Math. Soc. 147 (2019), 2321-2330
MSC (2010): Primary 11J91; Secondary 33E30
DOI: https://doi.org/10.1090/proc/14402
Published electronically: February 20, 2019
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Abstract: In this paper we define and study a filtration $ (\mathbf {G}_s)_{s\geq 0}$ on the algebra of values at algebraic points of analytic continuations of $ G$-functions: $ \mathbf {G}_s$ is the set of values at algebraic points in the disk of convergence of all $ G$-functions $ \sum _{n=0}^\infty a_n z^n$ for which there exist some positive integers $ b$ and $ c$ such that $ d_{bn}^{s} c^{n+1} a_n $ is an algebraic integer for any $ n $, where $ d_n =$$ \text {lcm}(1,2,\ldots ,n)$.

We study the situation at the boundary of the disk of convergence, and using transfer results from analysis of singularities we deduce that constants in $ \mathbf {G}_s$ appear in the asymptotic expansion of such a sequence $ (a_n)$.


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Additional Information

S. Fischler
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

T. Rivoal
Affiliation: Institut Fourier, CNRS et Université Grenoble 1, 100 rue des maths, BP 74, 38402 St Martin d’Hères Cedex, France

DOI: https://doi.org/10.1090/proc/14402
Keywords: $G$-functions, $G$-values, singularity analysis
Received by editor(s): February 8, 2018
Received by editor(s) in revised form: September 18, 2018
Published electronically: February 20, 2019
Additional Notes: Both authors have been supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French Program Investissement d’Avenir
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2019 American Mathematical Society