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Algebro-analytical study on special functions appeared in number theory

数论中出现的特殊函数的代数分析研究

基本信息

批准号：
15540050
负责人：
UENO Kimio
金额：
$ 1.6万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

First I tried to make correspondence, via Mellin transforms and inverse Mellin transforms, between linear relations held among multiple zeta values and connection relations held among multiple polylogarithms of one variable. I was succeeded in clarifying the correspondence between Ohno relations for multiple zeta values and the connection formula for multiple polylogarithms with respect to z→z-1. This result was published as the paper "Relations for Multiple Zeta Values and Mellin Transforms of Multiple Polylogarithms, Publ. RIMS, Kyoto Univ. 40 (2004), 537-564".Motivated by this research, I tried to study symmetry of the formal Knizhinik-Zamolodchikov(KZ) equation of one variable, in which the coefficients of the equation are regarded as noncommutative variables. Here the symmetry of the equation means transformation theory of fundamental solutions normalized at the singular points. The most universal generating function of multiple polylogarithms is a fundamental solution normalized … More at z=0. The connection matrices of these fundamental solutions are given by the so called Drinfeld associator, and the connection relations yield the duality relation and the hexagon relation for the Drinfeld associator. This result was published as the paper "The Sum Formula of Multiple Zeta Values and Connection Problem of the Formal Knizhinik-Zamolodchikov Equation, Development in Mathematics 14, Zeta Functions, Topology and Quantum Physics ed. Be T. Aoki et al., Springer (2005),145-170.Furthermore, I have been trying to generalize the symmetry theory in the formal KZ equation of many variables. I expect that, from this theory, the connection formulas for multiple polylogarithms of many variables. So far, I established the decomposition theorem, and the analycity theorem for normalized fundamental solutions of the KZ equation, from which one can deduce the pentagon relation for the Drinfeld associator. I proposed a conjecture that the harmonic product relations(or the series shuffle relations) of multiple polylogarithms are equivalent to the decomposition theorem. This result is announced in an invited project lecture in the autumn conference of MSJ at Osaka City University, 2006 September. Less

首先，我尝试通过梅林变换和梅林逆变换，在多个 zeta 值之间的线性关系和一个变量的多个多对数之间的连接关系之间建立对应关系。我成功地阐明了多个 zeta 值的大野关系与 z→z-1 的多个多对数的连接公式之间的对应关系。该结果发表为论文“Relations for Multiple Zeta Values and Mellin Transforms of Multiple Polylogarithms, Publ. RIMS, Kenya Univ. 40 (2004), 537-564”。受这项研究的启发，我尝试研究单变量形式 Knizhinik-Zamolodchikov(KZ) 方程的对称性，其中方程的系数被视为非交换变量。这里方程的对称性意味着奇点处归一化的基本解的变换理论。多重多对数最通用的生成函数是在 z=0 处归一化的基本解。这些基本解的连接矩阵由所谓的德林菲尔德关联器给出，并且连接关系产生了德林菲尔德关联器的对偶关系和六边形关系。这一结果发表为论文“The Sum Formula of Multiple Zeta Values and Connection Problem of the Formal Knizhinik-Zamolodchikov Equation, Development in Mathematics 14, Zeta Functions, Topology and Quantum Chemistry ed. Be T. Aoki et al., Springer (2005),145-170。此外，我一直在尝试将对称理论推广到许多形式的 KZ 方程中。变量。我期望从这个理论中得出许多变量的多个多对数的连接公式。到目前为止，我建立了分解定理，以及KZ方程归一化基本解的解析性定理，从中可以推导出Drinfeld关联器的五边形关系。我提出了一个猜想，多个多对数的调和乘积关系（或级数洗牌关系）是等价于分解定理。这一结果是在 2006 年 9 月大阪市立大学 MSJ 秋季会议的邀请项目演讲中宣布的。少