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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.

在本研究的开始，我的目的是研究的只是一个很自然的和唯一的模拟椭圆函数理论中的通常的n-乘法公式的分子。这种类比与经典的阿贝尔函数理论完全不同。我们的新n乘公式也是一个函数的有理函数，这与经典理论中的公式本质上是多元的相反，然而，在第二年的研究中，我发现了一个显著的分母行列式表达式。第二个研究年度也致力于对这一新表达的调查。亏格2的结果发表在《格拉斯哥数学杂志》上。（2002），以及一个关于属3的情况将发表在东京数学杂志上。一般属情况的结果也提交了，可以从网页上获得，许多研究人员正在下载它。此外，我受到爱丁堡数学学会的邀请，在第三年的后期，我对上述行列式表达式中出现的函数进行了数论研究。也就是说，关于罗朗系数的发展在原点的功能。系数与伯努利数（函数1/tan（u）的系数）和赫维茨数（分圆型椭圆函数p（u）的系数）非常相似。它们确实满足von Staudt-Clausen型定理和库默型同余关系。这些性质已被完全证明，该论文现已在网上发布。该补助金主要用于首席研究员和助理研究员的旅行，旨在寻找参考书目并在几个机构中展示结果。