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Abstract
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-functions are
power series in
solutions of linear differential equations, and whose Taylor
coefficients satisfy certain (non-)archimedean growth conditions. In
1929, Siegel proved that every generalized hypergeometric series
with rational parameters
is a
-function,
but rationality of parameters is in fact not necessary for a hypergeometric series to be a
-function.
In 1981, Galochkin found necessary and sufficient conditions on the parameters of a
series to be a
nonpolynomial
-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
-function,
which is Fuchsian with rational exponents.
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Keywords
$G$-functions, generalized hypergeometric series, Fuchsian
differential equations
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Mathematical Subject Classification
Primary: 33C20
Secondary: 11J91, 34M15
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Milestones
Received: 3 June 2021
Accepted: 26 July 2021
Published: 30 March 2022
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