Linear independence of values of G-functions

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 $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $\alpha$ such that $0 0$ 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 an 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 Pad\'e-type approximants. It makes use of results of Andr\'e, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion \`a la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.