Linear independence of values of $G$-functions

Abstract

Given any non-polynomial $G$-function $F(z)={\sum }_{k=0}^{\infty }{A}_k{z}^k$ of radius of convergence $R$, we consider the $G$-functions $F_n^{[s]}(z)={\sum }_{k=0}^{\infty }\frac{A_k}{(k+n{)}^s}{z}^k$ for any integers $s\ge 0$ and $n\ge 1$. For any fixed algebraic number $\alpha$ such that $0<|\alpha |<R$ and any number field $\mathbb{K}$ containing $\alpha$ and the $A_k$'s, we define ${\Phi }_{\alpha ,S}$ as the $\mathbb{K}$-vector space generated by the values $F_n^{[s]}(\alpha )$, $n\ge 1$ and $0\le s\le S$. We prove that $u_{\mathbb{K},F}\log (S)\le {\dim }_{\mathbb{K}}({\Phi }_{\alpha ,S})\le v_{F}S$ for any $S$, with effective constants $u_{\mathbb{K},F}>0$ and $v_F>0$, and that the family $(F_n^{[s]}(\alpha ){)}_{1\le n\le v_F,s\ge 0}$ 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.