Abstract

$G$-functions are power series in $\overline{\mathbb{Q}}[[z]]$ solutions of linear differential equations, and whose Taylor coefficients satisfy certain (non-)archimedean growth conditions. In 1929, Siegel proved that every generalized hypergeometric series ${}_{q+1}{F}_q$ 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 ${}_{q+1}{F}_q$ 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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