On the linear independence of values of $G$-functions

We consider a $G$-function $F(z)=\sum_{k=0}^{\infty} A_k z^k \in \mathbb{K}[[z]]$, where $\mathbb{K}$ is a number field, of radius of convergence $R$ and annihilated by the $G$-operator $L \in \mathbb{K}(z)[\mathrm{d}/\mathrm{d}z]$, and a parameter $\beta \in \mathbb{Q} \setminus \mathbb{Z}_{\leqslant 0}$. We define a family of $G$-functions $F_{\beta,n}^{[s]}(z)=\sum_{k=0}^{\infty} \frac{A_k}{(k+\beta+n)^s} z^{k+n}$ indexed by the integers $s$ and $n$. Fix $\alpha \in \mathbb{K}^* \cap D(0,R)$. Let $\Phi_{\alpha,\beta,S}$ be the $\mathbb{K}$-vector space generated by the values $F_{\beta,n}^{[s]}(\alpha)$, $n \in \mathbb{N}$, $0 \leqslant s \leqslant S$. We show that there exist some positive constants $u_{\mathbb{K},F,\beta}$ and $v_{F,\beta}$ such that $u_{\mathbb{K},F,\beta} \log(S) \leqslant \dim_{\mathbb{K}} \Phi_{\alpha,\beta,S} \leqslant v_{F,\beta} S$. This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case $\beta=0$. Our proof is an adaptation of their article "Linear independence of values of $G$-functions'' ([FR]), making use of the Andr\'e-Chudnovsky-Katz Theorem on the structure of the $G$-operators and of the saddle point method.