On the values of G-functions

TL;DR

This paper explores the properties of G-functions, showing that their values at algebraic points can be expressed as G-functions with rational coefficients, and characterizes quotients of such values as limits of ratios of G-function-generated sequences.

Contribution

It demonstrates that G-function values at algebraic points can be represented as G-functions with rational coefficients and large convergence radii, and characterizes quotients of these values as limits of G-function-based sequences.

Findings

Values of G-functions at algebraic points can be expressed as G-functions with rational coefficients.

Quotients of such G-function values are limits of ratios of G-function-generated sequences.

Provides a framework for irrationality proofs similar to Apéry's approach for zeta(3).

Abstract

Let f be a G-function (in the sense of Siegel), and x be an algebraic number; assume that the value f(x) is a real number. As a special case of a more general result, we show that f(x) can be written as g(1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a(n)/b(n), where the generating series of both sequences are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apery for zeta(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].