English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article, we show a new general linear independence criterion relatedto values of $G$-functions, including the linear independence of values atalgebraic points of contiguous hypergeometric functions, which is not knownbefore. 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 $(1\le k \ler, 1\le j \le r-1)$. 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, weprove 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 \ler-1)\end{align*} and $1$ are linearly independent over $K$. The essentialingredient is our term-wise formal construction of type II of Pad\'eapproximants together with new non-vanishing argument for the generalizedWronskian.

Generalized hypergeometric $G$-functions take linear independent values