Laurent type expansion of multiple polylogarithms at integer points

TL;DR

This paper investigates the local behavior of multiple polylogarithm functions at integer points by developing Laurent-type expansions involving power series and rational functions, with coefficients linked to regularized values.

Contribution

It introduces a Laurent expansion framework for multiple polylogarithms at integers, connecting coefficients to regularized values and enhancing understanding of their local properties.

Findings

Laurent expansions at integer points derived

Coefficients relate to regularized multiple polylogarithm values

Provides new tools for analyzing polylogarithm behavior near integers

Abstract

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational functions. The coefficients of these power series are the regularised values of the multiple polylogarithm functions at certain related integer points.