Identities for the multiple zeta (star) values

TL;DR

This paper establishes new identities for multiple zeta and zeta star values of arbitrary depth using integral methods, enabling polynomial expressions and evaluations of sum formulas involving these special values.

Contribution

It introduces novel identities for multiple zeta and star values, expressing certain sequences in terms of zeta values, polylogarithms, and logarithms, expanding the analytical tools for these constants.

Findings

Polynomial expressions for specific multiple zeta star values.

Evaluation of restricted sum formulas involving multiple zeta values.

New identities derived via integral and iterated integral methods.

Abstract

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying these formulas, we can prove that multiple zeta star values whose indices are the sequences $(\bar 1,\{1\}_m,\bar 1)$ and $(2,\{1\}_m,\bar 1)$ can be expressed polynomially in terms of zeta values, polylogarithms and $\ln2$. Finally, we also evaluate several restricted sum formulas involving multiple zeta values.