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A note on G-operators of order 2

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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https://hal.archives-ouvertes.fr/hal-03065680
Contributor : Tanguy Rivoal <>
Submitted on : Monday, December 14, 2020 - 9:36:25 PM
Last modification on : Monday, December 28, 2020 - 11:18:53 AM
Long-term archiving on: : Monday, March 15, 2021 - 8:24:18 PM

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Stéphane Fischler, Tanguy Rivoal. A note on G-operators of order 2. 2020. ⟨hal-03065680⟩

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