A discretization of the iterated integral expression of the multiple polylogarithm

TL;DR

This paper extends the discretization phenomenon of iterated integral expressions from multiple zeta values to multiple polylogarithms, providing two proofs and exploring applications.

Contribution

It introduces a discretization of the iterated integral expression for multiple polylogarithms, expanding on previous results for multiple zeta values with two distinct proofs.

Findings

Discretization of multiple polylogarithms' integral expressions

Two proofs: connected sums and induction on difference equations

Applications of the discretization phenomenon

Abstract

Recently, Maesaka, Watanabe, and the third author discovered a phenomenon where the iterated integral expressions of multiple zeta values become discretized. In this paper, we extend their result to the case of multiple polylogarithms and provide two proofs. The first proof uses the method of connected sums, while the second employs induction based on the difference equations that discrete multiple polylogarithms satisfy. We also investigate several applications of our main result.