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

Volume 170 / 2022

S. Fischler, T. Rivoal Colloquium Mathematicum 170 (2022), 321-340 MSC: Primary 33E20; Secondary 34M15, 34M35, 11J91. DOI: 10.4064/cm8600-3-2022 Published online: 4 July 2022


It is known that $G$-functions solving a linear differential equation of order $1$ with coefficients in $\overline {\mathbb Q}(z)$ are algebraic (and 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 solving 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 holonomic Nilsson–Gevrey arithmetic series of order 0 that encompass $G$-functions.


  • S. FischlerUniversité Paris-Saclay, CNRS
    Laboratoire de mathématiques d’Orsay
    91405 Orsay, France
  • T. RivoalInstitut Fourier, CNRS
    et Université Grenoble Alpes
    CS 40700
    38058 Grenoble Cedex 9, France

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