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Special Function of Several Variables

Research Project

Project/Area Number 04302005
Research Category

Grant-in-Aid for Co-operative Research (A)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionKobe University

Principal Investigator

TAKANO Kyoichi  Kobe Univ., Dep. of Math., Prof., 理学部, 教授 (10011678)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Masaaki  Kyushu Univ., Dep. of Math., Prof., 理学部, 教授 (30030787)
MAJIMA Hideyuki  Ochanomizu Univ., Dep. of Math., Prof., 理学部, 教授 (50111456)
KOHNO Mitsuhiko  Kumamoto Univ., Dep. of Math., Prof., 理学部, 教授 (30027370)
KIMURA Hironobu  Univ. of Tokyo, Dep. of Math., Asso. Prof., 大学院・数理科学研究科, 助教授 (40161575)
OKAMOTO Kazuo  Univ. of Tokyo, Dep. of Math., Prof., 大学院・数理科学研究科, 教授 (40011720)
Project Period (FY) 1992 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥7,600,000 (Direct Cost: ¥7,600,000)
Fiscal Year 1993: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1992: ¥4,300,000 (Direct Cost: ¥4,300,000)
Keywordshypergeometric 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
Research Abstract

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.

Report

(3 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • Research Products

    (40 results)

All Other

All Publications (40 results)

  • [Publications] 木村弘信: "On confluences of the general hypergeometric systems" Proc.of the Japan Academy. 69. 99-104 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 原岡喜重: "Contiguity relations of generalized confluent hypergeometric functions" Proc.of the Japan Academy. 69. 105-110 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 下村俊: "Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function Ф_2,I" Proc.Roy.Soc.Edinburgh. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 岩崎克則: "Twisted homology of the configulation spaces of n-points with application to the hypergeometric functions" Preprint Series:University of Tokyo,Department of Mathematical Sciences. 94-11. 1-68 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 喜多通武: "On hypergeometric functions in several variables II,The Wronskian of the hypergeometric functions" J.Math.Soc.Japan. 45. 645-669 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 松本圭司: "Monodromy of the hypergeometric differential equation of type(3,6)I" Duke Math.J.71. 403-426 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 高野恭一: "常微分方程式" 朝倉書店, 210 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Aizawa, K.and Tmita, Y.: "Unbounded viscosity solutions of fully nonlinear elliptic equations in R^n" Adv. Math. Sci. Appl.2. 297-316 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Gerard, R.and Tahara, H.: "Solutions holomorphes et singulieres d'equations aux derivees partielles sigulieres nonlineaires" Publ. RIMS, Kyoto Univ.29. 121-151 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Haraoka, Y.and Kimura, H.: "Contiguity relations of generalized confluent hypergeometric functions" Proc. Japan Acad. 69. 105-110 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Iwasaki, K.: "On a dynamical system on the matrix algebra, preprint series" Univ. of Tokyo. 94-3. 1-45 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Iwasaki, K.and Kita, M.: "Exterior power structure of the twisted de Rham cohomology associated to arrangements of hyperplanes in general position, preprint series" Univ. of Tokyo. 94-2. 1-26 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Iwasaki, K.and Kita, M.: "Twisted homology of the configulation spaces of n-points with application to the hypergeometric functions" preprint series, Univ. of Tokyo. 94-11. 1-68 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kamimura, Y.: "An inverse problem in bifurcation theory" J.Differential Equations. 106. 10-26 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kamimura, Y.: "An inverse problem in bifurcation theory II" J.Math. Soc. Japan.46. 89-110 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kawai, T.and Takei, Y.: "On the structure of Painleve transcendents with a large parameter" Proc. Japan Acad.69. 224-229 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kimura, H., Haraoka, Y.and Takano, K.: "The generalized confluent hypergeometric functions" Proc. Japan Acad.68. 290-295 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kimura, H., Haraoka, Y.and Takano, K.: "On confluence of the general hypergeometric systems" Proc. Japan Acad.69. 99-10 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Kita, M.: "On hypergeometric functions in several variables I.New integral representaions of Euler type" Japan. J.Math. 18. 25-74 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] KIta, M.: "On hypergeometric functions in several variables II.The Wronskian of the hypergeometric functions." J.Math. Soc. Japan. 45. 645-669 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Majima, H.: "Stokes structures of the confluent hypergeometric differential equations in two variables and resurgent equations, preprint" (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 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)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Matsumoto, K., Sasaki, T., Takayama, N.and Yoshida, M.: "Monodromy of the hypergeometric differential equation of type (3, 6) (II)20GD16 : to appear in Ann. Scuola Norm Sup. Pisa"

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Matsumoto, K.and Yoshida, M.: "Configulation space of 8 points on the projective line and a 5-dimensional Picard modular group" Compositio Math.86. 265-280 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Miyake, M.: "Relations of equations of Euler, Hermite and Weber via the heat equation" Funkcial. Ekvak.36. 251-273 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Shimomura, S.: "Asymptotic expansions and Stokes multiplies of the confluent hypergeometric function PHI_2, I." Proc. Roy. Soc. Edinburgh.123. (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] Terasoma, T.and Saito, T.: "A determinant formula for period integrals" Proc. Japan Acad.69. 131-135 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 木村弘信: "On confluences of the general hypergeometric systems" Proc.Japan Acad.69. 99-104 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 原岡喜重: "Contiguity relations of generalized confluent hypergeometric functions" Proc.Japan Acad.69. 105-110 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 下村俊: "Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function Φ_2 I" Proc.Roy.Soc.Edinburgh. (to appear).

    • Related Report
      1993 Annual Research Report
  • [Publications] 岩崎克則: "Twisted homology of the configulation spaces of n-points with application to hypergeometric functions" Preprint Series:Univ.Tokyo,Dept.math.94-11. 1-68 (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] 喜多通武: "On hypergeometric functions in several variables II,The Wronskian of the hypergeometric functions" J.Math.Soc.Japan. 45. 645-669 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 松本圭司: "Monodromy of the hypergeometric differential equation of type(3,6)I" Duke Math.J.71. 403-426 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 高野恭一: "常微分方程式" 朝倉書店, 210 (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] 木村 弘信: "The generalized confluent hypergeometric functions" Proc.Japan Acad.68. 290-295 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 木村 弘信: "On confluences of the general hypergeometric systems" Proc.Japan Acad.

    • Related Report
      1992 Annual Research Report
  • [Publications] 松本 圭司: "The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type(3,6)" International J.Math.3. 1-164 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 松本 圭司: "Monodromy of the hypergeometrid differential equations of type(k,n),I" Duke Math.J.

    • Related Report
      1992 Annual Research Report
  • [Publications] 三宅 正武: "Newton polygons and Gevrey indices for linear partial differential operators" Nagoya Math.J.128. 15-47 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 三宅 正武: "Relations of equations of Euler,Hermite and Weber via the heat equation" Funkcial.Ekvac.

    • Related Report
      1992 Annual Research Report

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Published: 1992-04-01   Modified: 2016-04-21  

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