Analytic continuation of multiple polylogarithms in positive characteristic

TL;DR

This paper develops a method for the analytic continuation of Carlitz multiple polylogarithms in positive characteristic using Artin-Schreier equations, introducing monodromy modules to analyze their branches and applying this to various properties.

Contribution

It introduces a novel continuation method for Carlitz polylogarithms in positive characteristic and explores their monodromy and functional relations.

Findings

Extended the domain of Carlitz multiple polylogarithms

Established monodromy module framework for branch analysis

Demonstrated branch independence of Eulerian properties

Abstract

Our aim of this paper is to propose a method of analytic continuation of Carlitz multiple (star) polylogarithms to the whole space by using Artin-Schreier equation and present a treatment of their branches by introducing the notion of monodromy modules. As applications of this method, we obtain (1) a method of continuation of the logarithms of higher tensor powers of Carlitz module, (2) the orthogonal property (Chang-Mishiba functional relations), (3) a branch independency of the Eulerian property.