Explicit evaluation of certain sums of multiple zeta-star values

TL;DR

This paper derives explicit formulas for sums of multiple zeta-star values with specific index patterns, revealing their rational parts and algebraic identities, advancing understanding of these special mathematical constants.

Contribution

It provides the first explicit formula for the rational part of certain multiple zeta-star value sums and interprets the results within harmonic algebra.

Findings

Explicit formula for the rational part of the sum

Interpretation as an identity in harmonic algebra

Extension of previous results on multiple zeta values

Abstract

Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of \pi. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.