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On the archimedean and nonarchimedean $q$-Gevrey orders

Author: Julien Roques
Journal: Proc. Amer. Math. Soc. 150 (2022), 1167-1176
MSC (2020): Primary 39A13
Published electronically: December 14, 2021
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Abstract: $q$-Difference equations appear in various contexts in mathematics and physics. The “basis” $q$ is sometimes a parameter, sometimes a fixed complex number. In both cases, one classically associates to any series solution of such equations its $q$-Gevrey order expressing the growth rate of its coefficients : a (nonarchimedean) $q^{-1}$-adic $q$-Gevrey order when $q$ is a parameter, an archimedean $q$-Gevrey order when $q$ is a fixed complex number. The objective of this paper is to relate these two $q$-Gevrey orders, which may seem unrelated at first glance as they express growth rates with respect to two very different norms. More precisely, let $f(q,z) \in \mathbb {C}(q)[[z]]$ be a series solution of a linear $q$-difference equation, where $q$ is a parameter, and assume that $f(q,z)$ can be specialized at some $q=q_{0} \in \mathbb {C}^{\times }$ of complex norm $>1$. On the one hand, the series $f(q,z)$ has a certain $q^{-1}$-adic $q$-Gevrey order $s_{q}$. On the other hand, the series $f(q_{0},z)$ has a certain archimedean $q_{0}$-Gevrey order $s_{q_{0}}$. We prove that $s_{q_{0}} \leq s_{q}$ “for most $q_{0}$”. In particular, this shows that if $f(q,z)$ has a nonzero (nonarchimedean) $q^{-1}$-adic radius of convergence, then $f(q_{0},z)$ has a nonzero archimedean radius converges “for most $q_{0}$”.

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Additional Information

Julien Roques
Affiliation: Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
MR Author ID: 803167
ORCID: 0000-0002-2450-9085

Keywords: Mahler equations
Received by editor(s): June 9, 2021
Published electronically: December 14, 2021
Additional Notes: This work was supported by the ANR De rerum natura project, grant ANR-19-CE40-0018 of the French Agence Nationale de la Recherche
Communicated by: Mourad Ismail
Article copyright: © Copyright 2021 by the author