Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualitéReport as inadecuate



 Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité


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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.



Author: Yves André

Source: https://archive.org/







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