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，我试图通过梅林转化和逆细蛋白逆变换来使对应关系，在多个Zeta值之间存在的线性关系与一个变量的多个小聚集体之间存在的线性关系。我成功地证明了多个Zeta值的OHNO关系与相对于Z→Z-1的多个多组载体的连接公式之间的对应关系。该结果被发表为“多种Zeta值和多个多聚类的梅林转换的论文，RIMS，京都大学40（2004），537-564”，这项研究对此进行了研究。非共同变量。在这里，方程的对称是指在单数点标准化的基本解决方案的转化理论。多种聚类的最通用的生成函数是对归一化的基本解决方案……在z = 0时更多。这些基本解决方案的连接矩阵由所谓的Drinfeld协会给出，并且连接关系产生了二元性关系和Drinfeld Associator的六边形关系。该结果作为“多个Zeta值的总和公式以及正式的Knizhinik-Zamolodchikov方程的总和公式，数学14中的发展，Zeta功能，拓扑，拓扑和量子物理学编辑。我希望，从这个理论中，我为迄今为止的多个变量的多个多组阶层建立了分解定理，以及用于KZ方程的归一化基本解决方案的分析理论，从中可以推导出五角菲尔德联想体的五角大楼关系（或者）与跨国公司相关的关系。分解定理在2006年9月的大阪市MSJ秋季会议上宣布了这一结果