论文标题
具有均匀和奇数求和索引的多个Zeta值的变体
Variants of Multiple Zeta Values with Even and Odd Summation Indices
论文作者
论文摘要
在本文中，我们定义和研究了2级的多个Zeta值的变体（称为多个混合值或多个$ m $ - 价值，简称MMV），这形成了交替多个Zeta值的空间的子空间。该变体包括霍夫曼的多个$ t $价值和kaneko-tsumura的多个$ t $价值作为特殊情况。我们设置了MMV的双层式关系（DBSF）的代数框架，并具有与普通的多个Zeta值相似的二元性，积分混合关系，串联弹药关系等的良好属性。此外，我们通过在这些$ t $ variants和kaneko-tsumura $ψ$ values之间建立一些明确的关系来研究Kaneko-Yamamoto类型多重Zeta值的几个$ t $ Variants。此外，我们证明所有Kaneko-tsumura $ψ$ - 可以通过使用多个相关的积分来用Kaneko-Tsumura多个$ t $ values表示，并为Kaneko-Tsumura $ψ$ VALUES找到一些双重性公式。我们还使用轮廓积分和残基定理的方法讨论了深度和三的MMV的明确评估。最后，我们研究了一些小重量的MMV子空间的尺寸。
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 $ψ$-values. Furthermore, we prove that all Kaneko-Tsumura $ψ$-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 $ψ$-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.
