On some multiple zeta-star values of one-two-three indices

TL;DR

This paper explores identities and dualities in multiple zeta-star values with specific index patterns, providing partial solutions to existing conjectures and extending known evaluations in the field of number theory.

Contribution

It introduces new identities and duality relations for multiple zeta-star values with indices formed by inserting 3 or 1 into sequences of 2s, advancing understanding of their structure.

Findings

Derived identities for multiple zeta-star values with specific indices.

Established duality relations among these values.

Provided partial solutions to existing conjectures in the field.

Abstract

In this paper, we present some identities for multiple zeta-star values with indices obtained by inserting 3 or 1 into the string 2,...,2. Our identities give analogues of Zagier's evaluation of \zeta(2,...,2,3,2,..., 2) and examples of a kind of duality of multiple zeta-star values. Moreover, their generalizations give partial solutions of conjectures proposed by Imatomi, Tanaka, Wakabayashi and the first author.