Explicit Relations of Some Variants of Convoluted Multiple Zeta Values

TL;DR

This paper explores explicit formulas and relations for convoluted variants of multiple zeta values and their level two analogs, using iterated integrals and extending known identities to new functions and representations.

Contribution

It introduces explicit formulas for convoluted multiple zeta values and their analogs, extending integral identities and graphical representations to these variants.

Findings

Derived explicit relations for convoluted MZVs and their alternating versions.

Extended integral-series identities to parametric MPLs and MMVs.

Generalized graphical representations to multiple-labeled posets.

Abstract

Kaneko and Yamamoto introduced a convoluted variant of multiple zeta values (MVZs) around 2016. In this paper, we will first establish some explicit formulas involving these values and their alternating version by using iterated integrals, which enable us to derive some explicit relations of the multiple polylogarithm (MPL) functions. Next, we define convoluted multiple $t$-values and multiple mixed values (MMVs) as level two analogs of convoluted MZVs, and, similar to convoluted MZVs, use iterated integrals to find some relations of these level two analogs. We will then consider the parametric MPLs and the parametric multiple harmonic (star) sums, and extend the Kaneko-Yamamoto's "integral-series" identity of MZVs to MPLs and MMVs. Finally, we will study multiple integrals of MPLs and MMVs by generalizing Yamamoto's graphical representations to multiple-labeled posets.