On variants of symmetric multiple zeta-star values and the cyclic sum formula

TL;DR

This paper explores variants of symmetric multiple zeta-star values, introduces regularization-based star analogues, and establishes a cyclic sum formula connecting these values with their p-adic counterparts.

Contribution

It defines star analogues of t-adic symmetric multiple zeta values using various regularizations and proves a cyclic sum formula linking these to p-adic finite multiple zeta-star values.

Findings

Defined star analogues via harmonic, shuffle, and Kaneko-Yamamoto regularizations.

Established the cyclic sum formula for t-adic symmetric multiple zeta-star values.

Connected cyclic sum formulas of t-adic symmetric and multiple zeta-star values.

Abstract

The $t$-adic symmetric multiple zeta values were defined Jarossay, which have been studied as a real analogue of $\boldsymbol{p}$-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko-Yamamoto's type regularization. We also present the cyclic sum formula for $t$-adic symmetric multiple zeta(-star) values, which is the counterpart of that for $\boldsymbol{p}$-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our new relationship that connects the cyclic sum formula for $t$-adic symmetric multiple zeta-star values and that for the multiple zeta-star values.