On $u$-Multiple Zeta Values in Positive Characteristic

TL;DR

This paper introduces $u$-multiple zeta values in positive characteristic, connecting them to classical and $q$-analog theories, and explores their properties and relations through the Carlitz module.

Contribution

It develops the theory of $u$-multiple zeta values in positive characteristic, establishing their limits, properties, and explicit relations, extending previous analogs.

Findings

Limits of finite multiple harmonic $u$-series yield Thakur's multiple zeta values.

Established explicit relations among Thakur's multiple zeta values.

Connected $u$-multiple zeta values to classical and $q$-analog theories.

Abstract

In this paper, we introduce the concepts of the $u$-bracket, finite multiple harmonic $u$-series, and $u$-multiple zeta values via the Carlitz module. These objects serve as function field counterparts to the classical theory of $q$-analogs. We prove that the "limits" of finite multiple harmonic $u$-series at Carlitz torsion points yield Thakur's multiple zeta values and finite multiple zeta values over $\mathbb{F}_r(\theta)$ from analytic and algebraic perspectives, respectively. This can be regarded as a positive characteristic analog of the results by Bachmann, Takeyama, and Tasaka [BTT18]. Furthermore, we investigate the properties of $u$-multiple zeta values and their expansions, obtaining a family of explicit relations among Thakur's multiple zeta values at both positive and non-positive indices.