The study of infinite product formulae for the Jackson integrals with Weyl group symmetry and their applications.

具有Weyl群对称性的Jackson积分的无限乘积公式及其应用的研究。

基本信息

批准号：
15540045
负责人：
ITO Masahiko
金额：
$ 2.37万
依托单位：
Aoyama Gakuin 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 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540045/
关键词：
Jackson integrals Weyl group symmetry asymptotic behavior elliptic theta functions Calogero model commuting differential operators 楕円データ関数

项目摘要

One of themes of the present research is to study the structure of an infinite product lattice. In order to obtain the infinite product expression we need two facts. One is the recurrence equation (two term relation) with respect to the parameters. The other is the principal term of its asymptotic behavior when we take the parameters to infinity. Once we have the recurrence relation, using it repeatedly and using the asymptotic behavior, we can immediately obtain the infinite product expression of the Jackson integral. But first of all, in order to carry this out, we need the recurrence relation itself and the explicit form of the principal term of asymptotic behavior. These two points are the difficult problem for the Jackson integral associated with the root systems. For these two points the systematical study had not been done yet until we developed the methods of calculating them.For the root system of type BCn, we eventually developed the simple and fundamental methods to obtain them as follows :1. The two terms of the recurrence relation are corresponding to some polynomials of degree 0 and degree n respectively. We introduced certain nice polynomials of middle degrees i such that 0<i<n. Using the Jackson integrals corresponding to these polynomials, we obtain the equations which interpolate the recurrence relation. We indicated the above procedure explicitly.2. Since the Jackson integral has many parameters, there are many choice of the direction for taking the parameters to infinity. We must choose a good direction from them if we calculate the asymptotic behavior. In the present research, we found a standard direction such that one can compute the asymptotic behavior in the very simple way.

当前研究的主题之一是研究无限乘积格的结构。为了获得无限的乘积表达式，我们需要两个事实。一是关于参数的递推方程（两项关系）。另一个是当我们将参数取为无穷大时其渐近行为的主项。一旦我们有了递推关系，反复使用它并利用渐近行为，我们就可以立即得到杰克逊积分的无限乘积表达式。但首先，为了实现这一点，我们需要递推关系本身和渐近行为主项的显式形式。这两点是与根系统相关的杰克逊积分的难题。对于这两点，我们还没有进行系统的研究，直到我们开发出了计算它们的方法。对于BCn型的根系，我们最终开发出了简单而基本的计算方法： 1．递推关系的两项分别对应0次和n次的一些多项式。我们引入了某些很好的中度 i 多项式，使得 0<i<n。使用与这些多项式相对应的杰克逊积分，我们获得了对递推关系进行插值的方程。我们明确指出了上述程序。2．由于杰克逊积分的参数较多，因此使参数趋于无穷大的方向有多种选择。如果我们计算渐近行为，我们必须从中选择一个好的方向。在目前的研究中，我们找到了一个标准方向，可以以非常简单的方式计算渐近行为。

项目成果

期刊论文数量（34）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Kenji TANIGUCHI: "On the symmetry of commuting differential operators with singularities along hyperplanes"International Mathematics Research Notices. To appear. (2004)

Kenji TANIGUCHI：“论沿超平面具有奇点的交换微分算子的对称性”国际数学研究通报。

DOI：
发表时间：
期刊：
影响因子：
0
作者：
通讯作者：

Askey-Wilson integrals associated with root systems

与根系统相关的 Askey-Wilson 积分

DOI：
发表时间：
期刊：
The Ramanujan Journal To appear
影响因子：
0
作者：
Kenji TNIGUCHI;Tomomi KAWAMURA;Kenji TANIGUCHI;Tomomi KAWAMURA;Masahiko ITO;Masahiko ITO;Masahiko ITO;Tomomi KAWAMURA;Masahiko ITO
通讯作者：
Masahiko ITO

Links and gordian number associated with certain generic immersions of circles

与某些通用的圆圈沉浸相关的链接和关键数字

DOI：
发表时间：
期刊：
Pacific Journal of Mathematics To appear
影响因子：
0
作者：
Kenji TNIGUCHI;Tomomi KAWAMURA;Kenji TANIGUCHI;Tomomi KAWAMURA;Masahiko ITO;Masahiko ITO;Masahiko ITO;Tomomi KAWAMURA
通讯作者：
Tomomi KAWAMURA

Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems