On the period interpretation for some special values of Thakur hypergeometric functions

TL;DR

This paper interprets special values of Thakur hypergeometric functions as periods of pre-$t$-motives, leading to new transcendence and independence results using advanced criteria, and extends these findings to related multiple polylogarithms.

Contribution

It provides a novel period interpretation of Thakur hypergeometric functions' special values, enabling new transcendence and independence results in positive characteristic.

Findings

Proved transcendence of certain special values.

Established linear independence among values.

Extended results to Kochubei multiple polylogarithms.

Abstract

In 1995, Thakur invented and studied positive characteristic analogues of hypergeometric functions. In this paper, we interpret the special values of those functions as periods of a pre-$t$-motive. As a consequence, we show their transcendence and linear independence results by using Chang's refined version of the Anderson-Brownawell-Papanikolas criterion. Furthermore, as by-products, we show some linear/algebraic independence results among the special values of Kochubei multiple polylogarithms according to our period interpretation and the corresponding refined criterion.