Discrete iterated integrals and cyclic sum formulas

TL;DR

This paper introduces a discrete analogue of iterated integrals using Riemann sums, establishes basic formulas, and proves cyclic sum formulas that extend to multiple polylogarithms.

Contribution

It develops a discrete framework for iterated integrals and proves cyclic sum formulas, connecting discrete and continuous cases in the context of multiple polylogarithms.

Findings

Established basic formulas for discrete iterated integrals.

Proved cyclic sum formulas for the discrete case.

Extended cyclic sum formulas to multiple polylogarithms.

Abstract

In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for discrete iterated integrals. They imply the cyclic sum formula for multiple polylogarithms.