# On the values of $G$-functions

### Stéphane Fischler

Université Paris-Sud 11, Orsay, France### Tanguy Rivoal

Université Grenoble I, Saint-Martin-D'hères Cedex, France

## Abstract

In this paper we study the set $G$ of values at algebraic points of analytic continuations of $G$-functions (in the sense of Siegel). This subring of $C$ contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions $_{p+1}F_{p}$ with rational coefficients. Its group of units contains non-zero algebraic numbers, $π$, $Γ(a/b)_{b}$ and $B(x,y)$ (with $a,b∈Z$ such that $a/b∈Z$, and $x,y∈Q$ such that $B(x,y)$ exists and is non-zero). We prove that for any $ξ∈G$, both $Reξ$ and $Imξ$ can be written as $f(1)$, where $f$ is a $G$-function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of $G∩R$ are exactly the numbers which can be written as limits of sequences $a_{n}/b_{n}$, where $∑_{n=0}a_{n}z_{n}$ and $∑_{n=0}b_{n}z_{n}$ are $G$-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for $ζ(3)$, 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.

## Cite this article

Stéphane Fischler, Tanguy Rivoal, On the values of $G$-functions. Comment. Math. Helv. 89 (2014), no. 2, pp. 313–341

DOI 10.4171/CMH/321