Special Functions in Many Variables

多变量中的特殊函数

• 批准号: 08454030
• 负责人: NOUMI Masatoshi
• 金额: $ 3.58万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454030/
• 关键词: hypergeometric function; Macdonald polynomial; spherical function; Grassmannian; confluent hypergeometric function; configuration space; Macdonald多項式; Grassmann多様体

The main subject of the this research project has been to construct a new prototype of the theory of special functions, by originating a systematic study of hypergeometric special functions in many variables. In this repect, the following results have been obtained from the viewpoints of (1) quantum group symmetry of difference systems, (2) confluent hypergeometric functions and Hamiltonian systems, (3) geometry of configurations spaces, and (4) representation theory and integral transformations, respectively.(1) M.Noumi developed a theory of sperical functions on quantum symmetric spaces in relation to quantum group symmetry. In terms of quantum groups, he gave a representation-theoretic realization of commuting families of q-difference operators and of the q-hypergeometric orthogonal polynomials of Macdonald type.(2) K.Takano studied in detail the procedure of confluence for hypergeometric functions over the Grassmannians, namely the degeneration of a general regular singularity to a confluent singularity. He also clarified the structure of the spaces of initial values and the mechanism of degeneration in Hamiltonian systems of Painleve' type.(3) T.Sasaki investigated the spaces of configurations of one nondegenerate quadratic hypersurface and n hyperplanes in the projective space. He determined the differential system for the associated hypergeometric integrals and described the symmetry of them. For the configurations in the projective plane, in particular, he consturcted explicit power series solutions and independent cycles, and clarified the relationship with Appell's hypergeometric functions.(4) From the viewpoint of Penrose transformations in symmetric domains, H.Sekiguchi studied the generalization of hypergeometric integrals and hypergeometric differential equations to higher ranks. She also established the finite dimensionality of their solution spaces by means of the method of unitary representations.

本研究项目的主要课题是通过对多变量超几何特殊函数的系统研究，构建特殊函数理论的新原型。在这方面，从（1）差分系统的量子群对称性，（2）汇合超几何函数和哈密顿系统，（3）构型空间的几何，以及（4）表示论和积分的角度得到了以下结果(1) M.Noumi 发展了与量子群对称相关的量子对称空间上的球面函数理论。在量子群方面，他给出了q-差算子的交换族和Macdonald型q-超几何正交多项式的表示论实现。(2) K.Takano详细研究了超几何函数的合流过程格拉斯曼主义者，即一般正则奇点退化为汇合奇点。他还阐明了Painleve型哈密顿系统的初值空间结构和退化机制。(3)T.Sasaki研究了射影空间中一个非退化二次超曲面和n个超平面的构型空间。他确定了相关超几何积分的微分系统并描述了它们的对称性。特别是对于射影平面上的构形，他构造了显式幂级数解和独立循环，并阐明了与Appell超几何函数的关系。(4)从对称域彭罗斯变换的角度，H.Sekiguchi研究了超几何积分和超几何微分方程到更高的等级。她还通过酉表示方法建立了它们的解空间的有限维性。

项目成果

Kimura,H.,Matumiya,A.and Takano,K.: "A normal form of Hamiltonian systems of several time variables with a regular singularity" J.Differential Equetions. 127-2. 337-364 (1996)

Kimura,H.、Matumiya,A. 和 Takano,K.：“具有正则奇点的多个时间变量的哈密顿系统的范式”J.微分方程。

Sasaki, T. and Yagi, T.: "Sectional curvature of projective invariant metrics on a strictly convex domain" Tokyo J. Math.19. 419-433 (1996)

Sasaki, T. 和 Yagi, T.：“严格凸域上射影不变度量的截面曲率”Tokyo J. Math.19。

M.Noumi, T.Umeda and M.Wakayama: "Dual pairs, spherical harmonics and a Capelli identity in quantum group threory" Compositio Mathematica. 104. 227-277 (1996)

M.Noumi、T.Umeda 和 M.Wakayama：“量子群理论中的对偶对、球谐函数和卡佩利恒等式”Compositio Mathematica。

K.Matsumoto and T.Sasaki: "On the system of differential equations associated with a quadric and hyperplanes" Kyushu J.Math.50. 493-131 (1996)

K.Matsumoto 和 T.Sasaki：“论与二次方程和超平面相关的微分方程组”九州 J.Math.50。

Noumi,M.: "Macdonald's symmetric polynomials as zonal spberical functions on some quantum homegeneous spaces" Advances in Math.123. 16-77 (1996)

Noumi,M.：“麦克唐纳的对称多项式作为某些量子同质空间上的区域球面函数”数学进展.123。