$G$-functions are power series in $mkern2muoverline{mkern-2mu mathbb{Q} mkern-2mu}mkern2mumkern-3muzmkern-3mu$ 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 Andr'e--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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