Linear independence of values of hypergeometric functions and arithmetic Gevrey series

TL;DR

This paper establishes new linear independence results for hypergeometric function values at multiple algebraic points over various number fields, using advanced Padé approximation techniques and non-vanishing arguments.

Contribution

It introduces a unified approach applying to all parameter regimes, extending previous results from single to multi-point values over general number fields.

Findings

Linear independence over algebraic number fields for hypergeometric values.

Extension of results from single-point to multi-point settings.

Validation of the method in complex and p-adic contexts.

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 Pad\'{e} 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 Pad\'{e} construction for families of contiguous hypergeometric values.