Variants of Multiple Zeta Values with Even and Odd Summation Indices

TL;DR

This paper introduces a new variant of multiple zeta values called MMVs, explores their algebraic properties, relations to existing values, explicit evaluations, and subspace dimensions, enriching the understanding of multiple zeta value structures.

Contribution

It defines MMVs, establishes their algebraic relations, connects them with known multiple zeta values, and investigates their explicit evaluations and subspace dimensions.

Findings

MMVs form a subspace of alternating multiple zeta values.

Explicit relations between T-variants and Kaneko-Tsumura ψ-values.

Dimensions of MMV subspaces are analyzed for small weights.

Abstract

In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study several $T$-variants of Kaneko-Yamamoto type multiple zeta values by establishing some explicit relations between these $T$-variants and Kaneko-Tsumura $\psi$-values. Furthermore, we prove that all Kaneko-Tsumura $\psi$-values can be expressed in terms of Kaneko-Tsumura multiple $T$-values by using multiple associated integrals, and find some duality formulas for Kaneko-Tsumura $\psi$-values. We also discuss the explicit evaluations for a kind of MMVs of depth two and three by using the method of contour integral and residue theorem. Finally, we investigate the dimensions of a few interesting subspaces of MMVs for small weights.