Alternating Multiple Mixed Values: Regularization, Special Values, Parity and Dimension Conjectures

TL;DR

This paper introduces and studies alternating multiple mixed values (AMMVs), a new variant of multiple zeta values of level four, revealing their algebraic properties, relations, and conjectured dimensions, with implications for understanding complex algebraic structures.

Contribution

The paper defines AMMVs, establishes their algebraic relations, proves a parity result, and conjectures their subspace dimensions, expanding the theory of multiple zeta values of level four.

Findings

AMMVs satisfy duality, shuffle, and stuffle relations.

Proved a parity conjecture for AMMVs.

Formulated conjectures on the dimensions of AMMV subspaces.

Abstract

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple $M$-values (AMMVs), forming a $\Q[i]$-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman's multiple $t$-values, Kaneko-Tsumura's multiple $T$-values, and the multiple $S$-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by us. We also investigate several alternating multiple $S$- and $T$-values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.