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。

• 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

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.

研究无限积格的结构是目前研究的主题之一。为了得到无穷积表达式，我们需要两个事实。一个是关于参数的递归方程（两项关系）。另一个是当参数趋于无穷时它的渐近特性的主项。一旦有了递归关系，重复使用它，利用渐近性质，我们可以立即得到Jackson积分的无穷积表达式。但首先，为了实现这个，我们需要递归关系本身和渐近行为的主项的显式形式。这两点是与根系有关的杰克逊积分的难题。对于这两点，在我们发展出计算方法之前，还没有作过系统的研究。对于BCn类型的根系，我们最终发展出简单而基本的获取方法如下：递归关系的两项分别对应于0次多项式和n次多项式。我们引入了一些中等次多项式，使得0<i<n。利用这些多项式对应的Jackson积分，我们得到了插值递推关系的方程。我们明确指出了上述程序。由于Jackson积分有很多参数，所以在取参数为无穷时，有很多方向的选择。如果要计算渐近行为，我们必须从中选择一个好的方向。在本研究中，我们找到了一个标准方向，使得人们可以用非常简单的方法来计算渐近行为。

项目成果

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。

Askey-Wilson integrals associated with root systems

与根系统相关的 Askey-Wilson 积分

• 期刊: The Ramanujan Journal To appear
• 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

• 期刊: Pacific Journal of Mathematics To appear
• 作者: Kenji TNIGUCHI;Tomomi KAWAMURA;Kenji TANIGUCHI;Tomomi KAWAMURA;Masahiko ITO;Masahiko ITO;Masahiko ITO;Tomomi KAWAMURA
• 通讯作者: Tomomi KAWAMURA

On the symmetry of commuting differential operators with singularities along hyperplanes

沿超平面具有奇点的交换微分算子的对称性

• 发表时间: 2004
• 期刊: International Mathematics Research Notices 36
• 作者: Kenji TNIGUCHI

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

• DOI: 10.1016/j.jat.2003.08.006
• 发表时间: 2003-10
• 期刊: J. Approx. Theory
• 作者: Masahiko Ito