Explicit Relations between Multiple Zeta Values and Related Variants

TL;DR

This paper derives new identities for multiple polylogarithms and harmonic sums, establishing explicit relations among various multiple zeta values and their variants through iterated integral methods.

Contribution

It introduces new identities and explicit relations between multiple zeta values, polylogarithms, and their variants, including a novel Apéry-type MZSV variant, using iterated integral techniques.

Findings

New identities for MPLs and MHSSs

Explicit relations between MZVs, MZSVs, and their variants

Introduction of MZBSVs and their connections to other zeta values

Abstract

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Ap\'{e}ry-type variant of MZSVs $\zeta^\star_B({\bf k})$ (called multiple zeta $B$-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.