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Research on products formulae for special values of Abehan functions

Abehan函数特殊值的乘积公式研究

基本信息

批准号：
12640004
负责人：
ONISHI Yoshihiro
金额：
$ 0.58万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

At the beginning of this reseach, I aimed to investigate on just the numerator of a quite natutal and unique analogy of the usual n-multiplication formula in elliptic function theory. This analogy is entirely different from the classical Abelian function theory. Our new n-multiplication formula is also a rational function of one function with contrary to the classical theory in which such formulae are essentailly of several variables.However, in the second research year, I found a remarkable determinantal expression of the denominator. The second research year is also devoted to investigation for this new expression. The result for the case of genus two was published in Glasgow Math. J.(2002), and one for the case of genus three will be publish in Tokyo J.Math. The result for the general genus case which was also submitted is available from a web page and many researchers are downloading it. Moreover I was invited from Edingburgh Math Soc., and discussed with several forign researchers.In the late of the third resaerch year, I made a number theoretical study for the functions appearing in the determinant expression above. Namely, about the Laurent coefficients of the developments at the origin of the functions. The coefficients resemble strongly to the Bernoulli numbers(the coefficients of the function 1/tan(u)), and the Hurwitz numbers, (the coefficients of an elliptic function p(u) of cyclotomic type). Indeed, they satisfy von Staudt-Clausen type theorem and Kummer type congruence relation. Such the properties were proved completely and the paper is now on the Web.This grant-aid is used mainly for the travels of the head and sub-investigators, with aimed at finding bibliographies and to present the results in several institutions.

在这项研究的开始，我的目的是研究椭圆函数理论中常见的n乘法公式的一个相当自然和唯一的类比的分子。这种类比与经典的阿贝尔函数理论完全不同。我们的新n乘法公式也是一个函数的有理函数，这与经典的n乘法公式本质上是多个变量的理论相反，但在第二个研究年，我发现了分母的一个显著的行列式表达式。第二个研究年也致力于对这一新表达的研究。第二类的研究结果发表在《格拉斯哥数学》杂志上。J.(2002)，关于属3的一个将在东京数学杂志上发表。也提交了一般属案例的结果，可以从网页上获得，许多研究人员正在下载它。第三年末，我对上述行列式中出现的函数进行了一些理论研究。即关于函数原点的发展的洛朗系数。这些系数与Bernoulli数(函数1/tan(U)的系数)和Hurwitz数(分圆型椭圆函数p(U)的系数)非常相似。事实上，它们满足von Staudt-Clausen型定理和Kummer型同余关系。这些性质得到了完全的证明，这篇论文现在已经上了网。这项赠款主要用于主管和副调查人员的旅行，目的是寻找书目，并在几个机构展示结果。