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Journal Articles Colloquium Mathematicum Year : 2022

A note on G-operators of order 2

Stéphane Fischler
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Tanguy Rivoal

Abstract

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, 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 $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions.
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Dates and versions

hal-03065680 , version 1 (14-12-2020)
hal-03065680 , version 2 (09-06-2021)

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Stéphane Fischler, Tanguy Rivoal. A note on G-operators of order 2. Colloquium Mathematicum, 2022, 170 (2), pp.321-340. ⟨10.4064/cm8600-3-2022⟩. ⟨hal-03065680v2⟩
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