Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras
Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras
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DOI：
10.1007/978-3-030-37031-2_10
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
Jianqiang Zhao
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文献类型：
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作者：
Jianqiang Zhao
近年来，多重Zeta值(MZV)得到了广泛的研究。目前，存在着数学家和物理学家定义和研究的几种不同类型的MZV(Q-MZV)类似物。在这篇文章中，我们通过考虑它们的双重洗牌关系(DBSF)和对偶关系，给出了这类eq-MZV的统一处理。其主要思想是对Castillo Medina等人使用的方法的修改和推广。到其他几种类型的Q-MZV，包括作者在2003年定义的一种。用不同的方法，竹山已经通过“正则化”研究了这种类型，并观察到存在一族新的线性关系，这些关系不是DBSFs的结果。我们在本文中称这些对偶关系为对偶关系，并将它们推广到所有其他类型的q-MZV。由于仍然存在一些缺失的关系，我们进一步定义了q-MZV的最一般类型，并定义了一种新的关系，称为-关系，用于进一步降低缺陷性。作为应用，我们将从理论上或数值上证实Okounkov关于权到12的某些q-MZV空间的维度的猜想。最后给出了一些相关的数值数据。
The multiple zeta values (MZVs) have been studied extensively in recent years. Currently there exist a few different types ofq-analogs of the MZVs (q-MZVs) defined and studied by mathematicians and physicists. In this paper, we give a uniform treatment of theseq-MZVs by considering their double shuffle relations (DBSFs) and duality relations. The main idea is a modification and generalization of the one used by Castillo Medina et al. to a few other types ofq-MZVs including the one defined by the author in 2003. With different approach, Takeyama already studied this type by “regularization” and observed that there exist a new family of-linear relations which are not consequences of the DBSFs. We call these duality relations in this paper and generalize them to all other types ofq-MZVs. Since there are still some missing relations we further define the most general type ofq-MZVs together with a new kind of relations called-relations which are used to lower the deficiencies further. As an application, we will confirm a conjecture of Okounkov on the dimensions of certainq-MZV spaces, either theoretically or numerically, for the weight up to 12. Some relevant numerical data are provided at the end.