Variants of Multiple Zeta Values with Even and Odd Summation Indices

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.

arXiv comment35 page, 1 graph

References in corpus (6)

• The Multiple Zeta Value Data Mine
• Multiple zeta values and Euler sums
• The parity theorem for multiple polylogarithms
• Multiple zeta-star values and multiple integrals
• Double Shuffle Relations of Double Zeta Values and Double Eisenstein Series of Level N
• Extensions of Euler Type Sums and Ramanujan Type Sums

Cited by in corpus (4)

• Apéry-Type Series and Colored Multiple Zeta Values
• Explicit Relations between Kaneko--Yamamoto Type Multiple Zeta Values and Related Variants
• Alternating Multiple Mixed Values: Regularization, Special Values, Parity and Dimension Conjectures
• Explicit Relations of Some Variants of Convoluted Multiple Zeta Values