Multivariable connected sums and multiple polylogarithms

TL;DR

This paper introduces multivariable connected sums, generalizes existing sums, and establishes identities that facilitate evaluating these sums and relating multiple polylogarithm values, including known relations like Ohno's.

Contribution

It presents the multivariable connected sum, proves a fundamental identity, and provides explicit evaluation procedures and relations among multiple polylogarithm values.

Findings

Established the fundamental identity for multivariable connected sums

Provided explicit evaluation procedures for these sums

Included relations such as Ohno's relations for multiple polylogarithms

Abstract

We introduce the multivariable connected sum which is a generalization of Seki-Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno's relations for multiple polylogarithms.