# Linear independence of values of $G$-functions

### Stéphane Fischler

Université Paris-Saclay, Orsay, France### Tanguy Rivoal

Université Grenoble Alpes, Grenoble, France

## Abstract

Given any non-polynomial $G$-function $F(z)=∑_{k=0}A_{k}z_{k}$ of radius of convergence $R$, we consider the $G$-functions $F_{n}(z)=∑_{k=0}(k+n)_{s}A_{k} z_{k}$ for any integers $s≥0$ and $n≥1$. For any fixed algebraic number $α$ such that $0<∣α∣<R$ and any number field $K$ containing $α$ and the $A_{k}$'s, we define $Φ_{α,S}$ as the $K$-vector space generated by the values $F_{n}(α)$, $n≥1$ and $0≤s≤S$. We prove that $u_{K,F}g(S)≤dim_{K}(Φ_{α,S})≤v_{F}S$ for any $S$, with effective constants $u_{K,F}>0$ and $v_{F}>0$, and that the family $(F_{n}(α))_{1≤n≤v_{F},s≥0}$ contains infinitely many irrational numbers. This theorem applies in particular when $F$ is a hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Padé-type approximants. It makes use of results of André, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion à la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.

## Cite this article

Stéphane Fischler, Tanguy Rivoal, Linear independence of values of $G$-functions. J. Eur. Math. Soc. 22 (2020), no. 5, pp. 1531–1576

DOI 10.4171/JEMS/950