Vol. 11, No. 1, 2022

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On Galochkin's characterization of hypergeometric $G$-functions

Tanguy Rivoal

Vol. 11 (2022), No. 1, 11–19

G-functions are power series in ¯[[z]] solutions of linear differential equations, and whose Taylor coefficients satisfy certain (non-)archimedean growth conditions. In 1929, Siegel proved that every generalized hypergeometric series q+1Fq with rational parameters is a G-function, but rationality of parameters is in fact not necessary for a hypergeometric series to be a G-function. In 1981, Galochkin found necessary and sufficient conditions on the parameters of a q+1Fq series to be a nonpolynomial G-function. His proof used specific tools in algebraic number theory to estimate the growth of the denominators of the Taylor coefficients of hypergeometric series with algebraic parameters. We give a different proof using methods from the theory of arithmetic differential equations, in particular the André–Chudnovsky–Katz theorem on the structure of the nonzero minimal differential equation satisfied by any given G-function, which is Fuchsian with rational exponents.

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$G$-functions, generalized hypergeometric series, Fuchsian differential equations
Mathematical Subject Classification
Primary: 33C20
Secondary: 11J91, 34M15
Received: 3 June 2021
Accepted: 26 July 2021
Published: 30 March 2022
Tanguy Rivoal
Institut Fourier
CNRS et Université Grenoble Alpes, CS 40700