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

Ce Xu, Lu Yan, Jianqiang Zhao

TL;DR

The paper develops alternating multiple mixed values (AMMVs) as a level-four, $\mathbb{Q}[i]$-linear extension of colored multiple zeta values, unifying alternating $t$-, $S$-, and $T$-type values under an iterated-integral framework. It establishes the algebraic structure via shuffle and stuffle products, derives regularized double shuffle relations, and proves a parity result that confirms prior conjectures. By exploiting arctangent-integral relations, it links AMSVs and AMTVs, and provides explicit evaluations for special cases. Weight-wise dimension analyses and conjectural pictures of subspace relations reveal rich and largely unexplored algebraic and geometric structures motivating further study of AMMVs and their motivic analogues.

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.

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

TL;DR

The paper develops alternating multiple mixed values (AMMVs) as a level-four, -linear extension of colored multiple zeta values, unifying alternating -, -, and -type values under an iterated-integral framework. It establishes the algebraic structure via shuffle and stuffle products, derives regularized double shuffle relations, and proves a parity result that confirms prior conjectures. By exploiting arctangent-integral relations, it links AMSVs and AMTVs, and provides explicit evaluations for special cases. Weight-wise dimension analyses and conjectural pictures of subspace relations reveal rich and largely unexplored algebraic and geometric structures motivating further study of AMMVs and their motivic analogues.

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 -values (AMMVs), forming a -subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman's multiple -values, Kaneko-Tsumura's multiple -values, and the multiple -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 - and -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.
Paper Structure (8 sections, 17 theorems, 154 equations, 4 tables)

This paper contains 8 sections, 17 theorems, 154 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Multiple mixed values and their alternating version
  3. Alternating multiple $S$/$T$-harmonic sums
  4. Main results
  5. Algebraic setup for AMMVs and finite double shuffle relations
  6. Regularization for divergent AMMVs and double shuffle relations
  7. Some results on AMSVs and AMTVs
  8. Dimension computation of AMMVs

Key Result

Proposition 2.2

The map ${\text{\em{M}}}:{\mathfrak{A}}_{\shuffle}^0\longrightarrow \mathbb{R}$ is an algebra homomorphism.

Theorems & Definitions (68)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 58 more