A generalized regularization theorem and Kawashima's relation for multiple zeta values

TL;DR

This paper introduces a new approach to Kawashima's relation among multiple zeta values by generalizing regularizations to Hurwitz type values and connecting it to the Kawashima function, avoiding Newton series.

Contribution

It presents a generalized regularization framework for Hurwitz type multiple zeta values and links it to Kawashima's relation through a novel method.

Findings

Established a generalized regularization theory for Hurwitz multiple zeta values

Proved a key property of the Kawashima function leading to Kawashima's relation

Provided a new proof of Kawashima's relation without Newton series

Abstract

Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values. Gaku Kawashima introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization of the theory of regularizations of divergent multiple zeta values to Hurwitz type multiple zeta values, and then relate it to the Kawashima function. Via this connection, we can prove a key property of the Kawashima function to obtain Kawashima's relation.