Séries Gevrey de type arithmétique，I.纯粹与二元理论
Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité
摘要：
Gevrey级数在分析中是普遍存在的;任何满足某些（可能是非线性的）解析微分方程的级数都是某个有理阶的Gevrey级数。本工作源于两个观察：1）经典的Gevrey级数，例如具有有理参数的广义超几何级数，享有阿基米德Gevrey条件的算术对应物; 2）出现在关于特殊函数的经典论文中的微分算子具有相当简单的结构：它们要么是Fuchsian的，要么只有两个奇点，0和无穷大，其中一个是规则的，另一个是不规则的，只有一个斜率。本文的主要思想是：算术性质1）解释了全局解析性质2）：在某一点上内射算术Gevrey解的存在性在很大程度上决定了多项式系数微分算子的全局行为。证明使用p-adic和复分析，以及对拉普拉斯变换的详细算法研究。
英文摘要：
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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