Algebro-analytical study on special functions appeared in number theory

首先，我试图通过Mellin变换和反Mellin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。我成功地澄清了多个zeta值的Ohno关系与关于z→z-1的多个多对数的连接公式之间的对应关系。这一结果发表在论文《多重多对数的多重Zeta值与Mellin变换的关系》中。京都大学学报，40(2004),537-564”。在此研究的激励下，我尝试研究一元形式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在奇点处归一化的基本解的变换理论。最普遍的多对数生成函数是一个基本解的归一化…更多在z=0处。这些基本解的连接矩阵由所谓的德林菲尔德关联子给出，这些连接关系产生了德林菲尔德关联子的对偶关系和六边形关系。这一结果发表在“形式Knizhinik-Zamolodchikov方程的多个Zeta值的和公式和连接问题”的论文中，数学发展14，Zeta函数，拓扑与量子物理主编。青木等，2010(2005),145-170。此外，我一直在尝试将对称理论推广到多变量形式KZ方程中。我希望，从这个理论中，可以得到多个变量的多重对数的连接公式。至此，我建立了KZ方程的分解定理和归一化基本解的解析定理，由此可以推导出德林菲尔德结合子的五边形关系。提出了一个猜想，即多个多对数的调和积关系（或级数洗牌关系）等价于分解定理。该结果于2006年9月在大阪城市大学举行的MSJ秋季会议上发表。少

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