Special Function of Several Variables

多变量的特殊函数

• 批准号: 04302005
• 负责人: TAKANO Kyoichi
• 金额: $ 4.86万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Co-operative Research (A)
• 财政年份: 1992
• 资助国家: 日本
• 起止时间: 1992 至 1993
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-04302005/
• 关键词: hypergeometric function; confluent HG function; twisted homology; twisted cohomology; confluce process; monodromy group; Stokes phenomenon; connection coefficient; モノドロミー群; 多変数超幾何関数; 多変数合流型超幾何関数; ストークス係数; ツイステッド・サイクル; 隣接関係式; 正準構造

1. On the study of (k, n) type hypergeometric functions (We abbreviate hypergeomentric as HF) : (1) The theory of twisted homologies and twisted cohomologies and interesection numbers was developed. We proposed a general framework based on the algebraic topology of the configulation spaces of n-points. A twisted simplicial theory and a twisted singular theory for the configulation spaces was developped and the exterior power structure of the homology module associated with the HG system of (3,6) type was obtained and all the cases where the group is of frute order were determined. (3) The relation between contiguous operators operators and the b functions for HG functions were obtained.2. On the study of(k, n)lambda type HG functions where lambda=(lambda_1, ・・・ lambda_1) is the decomposition of n : (1) The structure of the Lie algebra generated by contiguous operators for (k, n)lambda type HG functions was determined. (2) We found a limit process which reduces (k, n)lambda type HG functions (or system) to (k, n)mu type HG functions (or system) where mu is adjacent to lamdba. The process is just the confluence from Gauss to Kummer in the case where k = 2, n = 4 and lambda = (1,1,1,1), mu = (2,1,1,1). It acts well on Jodan groups, on characters and on differential equations.3. On asymptotic analysis : A general methpd of asymptotic for confluent HG functions was proposed. It consists of three parts : the first is Borel Transformation, the second is an analysis of the resurgent equation and the last is Laplace transformation. Based on the method, Stokes multipliers for the confluent HG two variables were calculated.

1.关于（k，n）型超几何函数（简称超几何中心函数HF）的研究：（1）发展了扭同调、扭上同调和交截数理论。我们提出了一个一般的框架的基础上的代数拓扑空间的n点。建立了代数空间的扭单纯性理论和扭奇异性理论，得到了（3，6）型HG系的同调模的外幂结构，并确定了群为f阶群的所有情形. (3)得到了HG函数的连续算子、算子与B函数之间的关系.关于（k，n）lambda型HG函数的研究，其中lambda=（λ da_1，···λ da_1）是n的分解：（1）确定了（k，n）lambda型HG函数由邻接算子生成的李代数的结构。(2)我们发现了一个极限过程，它将（k，n）λ型HG函数（或系统）约化为（k，n）μ型HG函数（或系统），其中μ与λ b相邻.当k = 2，n = 4，λ =（1，1，1，1），μ =（2，1，1，1）时，这个过程就是高斯到库默的汇流.它对Jodan群、特征标和微分方程都有很好的作用.在渐近分析方面：给出了合流HG函数渐近性的一般方法。它包括三个部分：第一部分是Borel变换，第二部分是对复苏方程的分析，第三部分是拉普拉斯变换。在此基础上，计算了合流HG二元的Stokes乘子。

项目成果

Matsumoto, K., Sasaki, T., Takayama, N.and Yoshida, M.: "Monodromy of the hypergeometric differential equation of type (3, 6) (I)" Duke Math. J.71. 403-426 (1993)

Matsumoto, K.、Sasaki, T.、Takayama, N. 和 Yoshida, M.：“(3, 6) (I) 型超几何微分方程的单律”杜克数学。

Gerard, R.and Tahara, H.: "Solutions holomorphes et singulieres d'equations aux derivees partielles sigulieres nonlineaires" Publ. RIMS, Kyoto Univ.29. 121-151 (1993)

Gerard, R. 和 Tahara, H.：“非线性部分派生部分的全态和奇异方程的解”Publ。

Kawai, T.and Takei, Y.: "On the structure of Painleve transcendents with a large parameter" Proc. Japan Acad.69. 224-229 (1993)

Kawai, T. 和 Takei, Y.：“论具有大参数的 Painleve 超越数的结构”Proc。

Kita, M.: "On hypergeometric functions in several variables I.New integral representaions of Euler type" Japan. J.Math. 18. 25-74 (1992)

Kita, M.：“论多变量中的超几何函数 I.欧拉型的新积分表示”，日本。

KIta, M.: "On hypergeometric functions in several variables II.The Wronskian of the hypergeometric functions." J.Math. Soc. Japan. 45. 645-669 (1993)

KIta, M.：“关于多变量的超几何函数 II。超几何函数的朗斯基式。”