On finite Carlitz multiple polylogarithms

TL;DR

This paper introduces finite Carlitz multiple polylogarithms and demonstrates their ability to express all finite multiple zeta values over rational function fields as linear combinations, aligning with existing Thakur MZV formulas.

Contribution

It defines finite Carlitz multiple polylogarithms and proves their sufficiency to represent all finite multiple zeta values over $ ext{F}_q( heta)$.

Findings

Finite Carlitz multiple polylogarithms are introduced.

All finite multiple zeta values over $ ext{F}_q( heta)$ can be expressed using these polylogarithms.

The results align with Thakur MZV formulas.

Abstract

In this paper, we define finite Carlitz multiple polylogarithms and show that every finite multiple zeta value over the rational function field $\mathbb{F}_{q}(\theta)$ is an $\mathbb{F}_{q}(\theta)$-linear combination of finite Carlitz multiple polylogarithms at integral points. It is completely compatible with the formula for Thakur MZV's established in [C14].