On Galochkin's characterization of hypergeometric \(G\)-functions (Q2121974)

\(G\)-functions are power series in \(\overline{\mathbb{Q}}[\mkern-3mu[z]\mkern-3mu]\) solutions of linear differential equations, whose Taylor coefficients satisfy specific (non-)archimedean growth conditions. In 1929, Siegel established that a generalized hypergeometric series with rational parameters, denoted by \(_{q+1}F_q\), is a \(G\)-function. However, rationality of parameters is not required for a hypergeometric series to be a \(G\)-function. Galochkin discovered in 1981 necessary and sufficient conditions on the parameters of a \(_{q+1}F_q\) series to be a non-polynomial \(G\)-function. He used in his proof specific tools from algebraic number theory to estimate the growth of the denominators of the Taylor coefficients of hypergeometric series. Here, a different proof is provided by the author using techniques from the theory of arithmetic differential equations, particularly 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.