Linear independence of values of hypergeometric functions and arithmetic Gevrey series
Sinnou David, Noriko Hirata-Kohno, Makoto Kawashima
TL;DR
The paper develops a universal Padé-approximation framework for generalized hypergeometric functions ${}_pF_q$ across all parameter regimes, and proves linear independence of their values at several algebraic points over arbitrary algebraic number fields by coupling type II Padé approximants with a novel non-vanishing Hermite-type generalized Wronskian. It treats the three main regimes ($p=q+1$, $p<q+1$, $p>q+1$) uniformly, extending prior one-point results to multi-point settings in both complex and $p$-adic contexts, and provides effective zero estimates and quantitative independence measures. The methodology unifies Euler-type, $G$- and $E$-functions via arithmetic Gevrey theory, enabling a single, flexible approach to families of contiguous hypergeometric values and delivering strengthened versions of results by Chudnovsky, Nesterenko, Sorokin, Delaygue, and others. The work has significant implications for transcendence and algebraic independence questions in hypergeometric contexts and demonstrates the power of Padé-type constructions in a broad arithmetic-differential setting.
Abstract
We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over arbitrary algebraic number fields. Our approach combines constructions of type II Padé approximants with a novel non-vanishing argument for generalized Wronskians of Hermite type. The method applies uniformly across all parameter regimes. Even for $p = q+1$, we extend known results from single-point to multi-point settings over general number fields, in the both complex and $p$-adic settings. When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting. Our results generalize and strengthen earlier work by Chudnovsky's, Nesterenko, Sorokin, Delaygue and others, and demonstrate the flexibility of our Padé construction for families of contiguous hypergeometric values.