A note on $G$-operators of order $2$

IF 0.4 4区 数学 Q4 MATHEMATICS Colloquium Mathematicum Pub Date : 2021-06-10 DOI:10.4064/cm8600-3-2022
S. Fischler, T. Rivoal
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引用次数: 4

Abstract

It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.
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关于$G$-阶为$2的算子的一个注记$
众所周知,系数为Q(z)的1阶线性微分方程的g函数解是代数的(具有非常精确的形式)。当阶数为2时,一般结果是未知的。本文确定了系数在Q(z)上的1阶非齐次方程的g函数解的形式,以及Q(z)上二阶微分阶的g函数f的解的形式,并且使得f和f '在C(z)上是代数相关的。我们的结果更普遍地适用于包含g函数的0阶完整Nilsson-Gevrey等差级数。
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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
61
审稿时长
6-12 weeks
期刊介绍: Colloquium Mathematicum is a journal devoted to the publication of original papers of moderate length addressed to a broad mathematical audience. It publishes results of original research, interesting new proofs of important theorems and research-expository papers in all fields of pure mathematics. Two issues constitute a volume, and at least four volumes are published each year.
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