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.

在这项研究的开头，我的目的是研究椭圆函数理论中常规n-义务公式的相当自然而独特的类比的分子。这种类比与经典的阿贝尔功能理论完全不同。我们的新n-义务公式也是一个函数的合理功能，与经典理论相反，这种公式在几个变量中是符合几个变量的经典理论。在第二个研究年中，我发现了分母的明显决定性表达。第二个研究年度还致力于调查这种新表达。二属属的结果发表在格拉斯哥数学中。 J.（2002），第三属的情况将在东京J.Math发表。一般属案件的结果也可以从网页上获得，许多研究人员正在下载它。此外，我受到了爱丁堡数学学院的邀请，并与几位前有限的研究人员进行了讨论。在第三年后期，我对上述决定性表达式出现的功能进行了许多理论研究。也就是说，关于功能起源的发展的劳伦斯系数。系数非常类似于Bernoulli数字（函数1/TAN（U）的系数）和Hurwitz数字（椭圆形函数P（U）的系数P（u）的系数P（u）的系数）。确实，他们满足了von staudt-clausen型定理和kummer型一致性关系。这样的属性已被完全证明，并且本文现在已在网络上。此授予AID主要用于头部和子注视者的旅行，旨在查找书目并在几个机构中介绍结果。