Shuffle product of finite multiple polylogarithms

TL;DR

This paper introduces a finite sum analogue of multiple polylogarithms, proves they satisfy a shuffle relation, and provides an algebraic interpretation of this product, extending the understanding of polylogarithm structures.

Contribution

It defines a finite sum analogue of multiple polylogarithms and proves a shuffle relation for them, offering new algebraic insights.

Findings

Finite sum analogue satisfies shuffle relation

Algebraic interpretation of the shuffle product

Extension of polylogarithm theory

Abstract

In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition due to Komori-Matsumoto-Tsumura. As a corollary, we give an algebraic interpretation of our shuffle product.