论文标题
交替多种混合价值：正则化，特殊值，奇偶校验和维度猜想
Alternating Multiple Mixed Values: Regularization, Special Values, Parity and Dimension Conjectures
论文作者
论文摘要
在本文中，我们定义和研究了第四级的多个Zeta值（MZV）的变体，称为交替多个混合值或交替的多个$ m $ - 价值（AMMV），形成了$ \ q [i] $ - 四级彩色MZV的子空间。该变体包括Hoffman的多个$ t $值的交替版本，Kaneko-Tsumura的多个$ t $值，以及以前由作者作为特殊案例研究的多个$ s $值。我们展示了与普通MZV相似的良好属性，例如广义二元性，整体式洗牌和系列材料关系。建立代数框架后，我们通过采用第四级的颜色MZV来得出AMMV的正规化双层式关系。作为一个重要的应用，我们证明了先前猜想的AMMV的奇偶校验结果。我们还通过建立涉及Arctangent功能的积分的一些明确关系来研究几个交替的多个$ S $ - 和$ t $ - 值。最后，我们计算了小于9的AMMV的一些有趣的子空间的维度。在理论和数值证据的支持下，由数值和符号计算的辅助证据支持，我们制定了一些关于AMMVS上述子空间的尺寸的猜想。这些猜想暗示了一些非常丰富但以前被忽略的代数和几何结构与这些矢量空间相关。
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.
