S\\éries Gevrey de type arithm\\étique, I. Th\\éor\èmes de puret\\é et de dualit\\é

Abstract : Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g. generalized hypergeometric series with rational parameters, enjoy arithmetic counterparts of the Archimedean Gevrey condition; 2) the differential operators which occur in classical treatises on special functions have a rather simple structure: they are either Fuchsian, or have only two singularities, 0 and infinity, one of them regular, the other irregular with a single slope... The main idea of the paper is that the arithmetic property 1) accounts for the global analytic property 2): the existence of an injective arithmetic Gevrey solution at one point determines to a large extent the global behaviour of a differential operator with polynomial coefficients. Proofs use both p-adic and complex analysis, and a detailed arithmetic study of the Laplace transform.
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Yves André. S\\éries Gevrey de type arithm\\étique, I. Th\\éor\èmes de puret\\é et de dualit\\é. Annals of Mathematics, Princeton University, Department of Mathematics, 2000, 151, pp.705-740. ⟨hal-00010037⟩



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