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Gevrey series of arithmetic type. I: Purity and duality theorems. (Séries Gevrey de type arithmétique. I: Théorèmes de pureté et de dualité.) (French) Zbl 1037.11049
Summary: Gevrey series are ubiquitous in analysis; any series satisfying some (possibly nonlinear) 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.
For Part II see ibid. 151, 741–756 (1990; Zbl 1037.11050).

##### MSC:
 11J81 Transcendence (general theory) 34M99 Ordinary differential equations in the complex domain
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