Toward a unified theory special functions of several variables

走向统一理论的多变量特殊函数

• 批准号: 08454033
• 负责人: KIMURA Hironobu
• 金额: $ 1.86万
• 依托单位: Department of Mathematics, Kumamoto 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-08454033/
• 关键词: special function; hypergeometric; de Rham theory; confluence; Lie algebra; regular element; de Rham cohomology; stratification; Grassmann多様体; 一般超幾何関数; Airy関数; homology; conomology; holonomic系; 鞍部点法

The objective of this project is to study the general hypergeometric functions (GHF) which were introduced by us to treat the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function.1 : GHFs are defined as solutions of certain holonomic systems on the Grassmannian Gr_<r, n> and they have the integral representations in a formal sense. We try to understand these integrals in the framework of the de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here we treat this problem in the particular cases of GHF,the case of generalized Airy functions and the case of GHFs given by the one dimensional integrals.(1)In relation to the problem of expressing the holonomic system for the generalized Airy functions as the integrable holonomic connection outside of the singlar locus, we computed in [3] the cohomology group of the rational twisted de Rham complex associated with … More the representation. We showed that the cohomology groups vanish except for the r-th one and that dim H^r=_<n-2>C_<r-1>. Moreover we presented the conjecture that a basis of H^r is given in thems of the Schur functions.(2)We understand domains of integrations for the generalized Airy integrals as cycles of a homology group on P^r with the family of supports defined by the integrand. By using the r-dimensional saddle point method, we showed in [4] that the homology groups are trivial except for the r-th one and that r-th homology group forms a local system of Z-modules on the space of independent variables of the functions rank _<n-2>C_<r-1>.(3)In the case where the GHFs are given by the one dimensional integrals (in other terms, the confluent case of Lauricella's F_D), we showed that the rational de Rham cohomology groups are trival except for H^1, and gave a basis of H^1 explicitly.2 : It is known that the other special functions of confluent type are derived from the Gauss hypergeometric function by the limit processes called confluences. In [5] we showed that this phenomenon can be explained by the adjacency relations among the strata of the stratification naturally introduced in the set of regular elements in the Lie algebra gl_n. Furthermore we generalized the above limit process to GHF in general. Less

本文的主要目的是研究广义超几何函数（GHF），它是我们为了处理Gauss超几何函数、库默合流超几何函数、Bessel函数、Hermite函数和Airy函数等经典特殊函数而引入的。1：GHF定义为Grassmannian Gr_<r，n>上某些完整系统的解，它们具有形式意义上的积分表示。我们试图理解这些积分的框架下的德拉姆理论，即作为对偶配对的上循环和循环的某些上同调和同调群。在这里，我们处理这个问题的特殊情况下的GHF，广义Airy函数的情况下，GHF的情况下，由一维积分。(1)In关于将广义Airy函数的完整系统表示为奇异轨迹外的可积完整联络的问题，我们在[3]中计算了与下列有关的有理扭曲de Rham复形的上同调群： ...更多信息 代表性。证明了除了第r个上同调群外，其余上同调群均为零，且dim H^r=_<n-2>C_<r-1>。并提出了H^r的一个基是以Schur函数形式给出的猜想。(2)We将广义艾里积分的积分域理解为P^r上同调群的圈，其支集族由被积函数定义。利用r维鞍点方法，我们在[4]中证明了除了第r个同调群外，其余同调群都是平凡的，并且第r个同调群构成函数rank _ C_的自变量空间上的局部Z-模系<n-2><r-1>。(3)In在GHF由一维积分给出的情况下（换句话说，Lauricella的F_D的汇合情况），我们证明了有理de Rham上同调群除了H^1之外都是平凡的，并明确给出了H^1的一个基。2：已知汇合型的其他特殊函数都是由Gauss超几何函数通过极限过程（称为汇合）导出的。在[5]中，我们证明了这一现象可以用李代数gl_n的正则元集合中自然引入的分层层之间的邻接关系来解释。进一步将上述极限过程推广到一般的广义HF。少

项目成果

H.Kimura: "On rational de Rham cohomology associated with the generalized Airy function, to appear in Annali di Scuola Norm." Sup.di Pisa. 24. (1997)

H.Kimura：“关于与广义艾里函数相关的有理德拉姆上同调，出现在 Annali di Scuola Norm 中。”

Hiroyoshi Yamaki: "A conjecture of Frobenius" Sugaku Exposition. 10・1. 69-85 (1996)

山木博吉：“弗罗贝尼乌斯的猜想”Sugaku Exposition 10・1（1996）。

木村 弘信, 小板橋 俊幸: "Normalizer of maximal abelian subgroups of GL(n) and general hypergeometric functions." Kumamoto J.Math.9. 13-43 (1996)

Hironobu Kimura、Toshiyuki Koitabashi：“GL(n) 的最大阿贝尔子群和一般超几何函数的归一化器。Kumamoto J.Math.9 (1996)”

Kotaro Yamada: "Surfaces of constant mean curvature c in H^3(-c^2)with prescribed hyperbolic Gauss map" Mathematische Annalen. 304. 203-224 (1996)

Kotaro Yamada：“具有规定的双曲高斯图的 H^3(-c^2) 中恒定平均曲率 c 的表面”数学年鉴。

Hironobu Kimura: "On the homology group associated with the generalized Airy function" Kumamoto Journal of Mathematics. 10. (1997)

Hironobu Kimura：“论与广义艾里函数相关的同调群”熊本数学杂志。