Modern development of special functions - approach from the representation theory and the integrals

特殊函数的现代发展 - 来自表示论和积分的方法

• 批准号: 09440020
• 负责人: MIMACHI Katsuhisa
• 金额: $ 6.27万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440020/
• 关键词: hypergeometric functions; integrals; representation theory; Macdonald polynomials; QKZ equation; Selberg integrals; de Rham theory; Painleve equation; de Rham理論; q-差分系; Hecke環; セルバーグ型積分

The purpose of the present research was to settle the viewpoint to unify the theory of hypergeometic function associated with the root system and the theory of integrals. Concrete theme of this work was the following : 1. De Rham theory (Study of homology and cohomology associated with Selberg type integrals, which appear as the spherical functions of A type), 2. Relationship between the representations of several kinds of algebras (Hecke algebras and so on) and the integrals, 3. Application to Painleve equations (special polynomials such as Okamoto polynomials), 4. Application to mathematical physics (Calogero system, correlation functions in conformal field theory or solvable lattice models). The results of the head investigator were mainly about 1 and 2, those of Hanamura were about 2, those of Noumi and Yamada were about 3. Matsui's help was valuable in the study of 4, Ochiai's in 2 and 4, Wakayama's in 2, Kato's in 1.Anyway, we have obtained a lot of results through the period of the present research project. As an evidence, many of papers had appeared in the journal of excellent level.

本研究的目的是解决与根系相关的超几何函数理论与积分理论统一的问题。具体的工作主题如下：1. De Rham理论（研究与Selberg型积分相关的同调和上同调，这些积分表现为A型球函数），2。几种代数（Hecke代数等）的表示与积分的关系。应用Painleve方程（特殊多项式，如冈本多项式），4。应用于数学物理（Calogero系统，共形场论或可解晶格模型中的相关函数）。首席调查员的结果主要是1和2左右，花村的结果是2左右，能见和山田的结果是3左右。松井的帮助在4的研究中是有价值的，落合的帮助在2和4的研究中是有价值的，和歌山的帮助在2的研究中是有价值的，加藤的帮助在1的研究中是有价值的。作为佐证，许多论文曾在优秀期刊上发表。

项目成果

K. Mimachi: "Askey-Wilson polynomials by means of a q-Selberg type integral"Adv. In Math.. 147. 315-327 (1999)

K. Mimachi：“利用 q-Selberg 型积分的 Askey-Wilson 多项式”Adv。

K.Mimachi: "Barnes type integral and the Meizner-Pollaczek polynomials"Lett,Math,Phys.. 48. 365-373 (1999)

K.Mimachi：“Barnes 型积分和 Meizner-Pollaczek 多项式”Lett,Math,Phys.. 48. 365-373 (1999)

K.Mimachi: "A new derivation of the inner product formula for the Macdonald symmetric polynomealss" Composit.Math.113. 117-122 (1998)

K.Mimachi：“麦克唐纳对称多项式内积公式的新推导”Composit.Math.113。

K. Mimachi: "A solution of the quantum Knizhnik Zamolodchikov equation of type CィイD2nィエD2"Commun. Math. Phys.. 197. 229-246 (1998)

K. Mimachi：“C2D2nD2 型量子 Knizhnik Zamolodchikov 方程的解”Commun Math。197. 229-246 (1998)

K.Mimachi: "A multidimensional generalization of the Barnes integral and the continuous Hahn polynomial"Jour,Math,Anal,and Appl,. 234. 67-76 (1999)

K.Mimachi：“巴恩斯积分和连续哈恩多项式的多维推广”Jour、Math、Anal 和 Appl，。