Explicit evaluation of certain sums of multiple zeta-star values

Abstract

Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence $(3,1,3,1,\ldots,3,1)$ 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.