Generalized hypergeometric $G$-functions take linear independent values

Abstract

In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let $K$ be any algebraic number field and $v$ be a place of $K$. Let $r\in\mathbb{Z}$ with $r\ge2$. Consider $a_1,\ldots,a_{r}, b_1,\ldots,b_{r-1}\in \mathbb{Q}\setminus\{0\}$ not being negative integers. Assume neither $a_k$ nor $a_k+1-b_j$ be strictly positive integers $(1\le k \le r, 1\le j \le r-1)$. Let $\alpha_1,\ldots,\alpha_m\in K\setminus\{0\}$ with $\alpha_1,\ldots,\alpha_m$ pairwise distinct. By choosing sufficiently large $\beta\in \mathbb{Z}$ depending on $K$ and $v$ such that the points $\alpha_1/\beta,\ldots,\alpha_m/\beta$ are closed enough to the origin, we prove that the $rm+1$ numbers~$:$ \begin{align*} &{}_{r}F_{r-1} \biggl(\begin{matrix} a_1,\ldots, a_r\\ b_1, \ldots, b_{r-1} \end{matrix} \biggm| \dfrac{\alpha_i}{\beta}\biggr)\enspace, \ \ {}_{r}F_{r-1} \biggl(\begin{matrix} a_1+1,\ldots,\ldots,\ldots,a_r+1\\ b_1+1, \ldots, b_{r-s}+1,b_{r-s+1},\ldots,b_{r-1} \end{matrix} \biggm| \dfrac{\alpha_i}{\beta}\biggr)\enspace\\ &(1\le i \le m, 1\le s \le r-1)\end{align*} and $1$ are linearly independent over $K$. The essential ingredient is our term-wise formal construction of type II of Pad\'e approximants together with new non-vanishing argument for the generalized Wronskian.