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

Abstract

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 $\beta \in \mathbb{Q}\setminus {\mathbb{Z}}_{⩽0}$. 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 $\alpha \in {\mathbb{K}}^{\ast }\cap D(0,R)$. Let ${\Phi }_{\alpha ,\beta ,S}$ be the $\mathbb{K}$-vector space generated by the values $F_{\beta ,n}^{[s]}(\alpha )$, $n\in \mathbb{N}$, $0⩽s⩽S$. We show that there exist some positive constants $u_{\mathbb{K},F,\beta }$ and $v_{F,\beta }$ such that $u_{\mathbb{K},F,\beta }\log (S)⩽{\dim }_{\mathbb{K}}{\Phi }_{\alpha ,\beta ,S}⩽v_{F,\beta }S$. 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 André-Chudnovsky-Katz Theorem on the structure of the $G$-operators and of the saddle point method.

References in corpus (1)

• Quantitative problems on the size of $G$-operators

Cited by in corpus (1)

• Quantitative problems on the size of $G$-operators