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Toward a unified theory of special functions of several variables

Research Project

Project/Area Number 09640205
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionDepartment of Mathematics, Kumamoto University

Principal Investigator

KIMURA Hironobu  Department of Mathematics, Kumamoto University, Professor of Mathematics, 理学部, 教授 (40161575)

Co-Investigator(Kenkyū-buntansha) OKAMOTO Kazuo  University of Tokyo, Graduate school of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40011720)
YAMADA Kotaro  Kumamoto University, Department of Mathematics, Associate Professor, 理学部, 助教授 (10221657)
HARAOKA Yoshishige  Kumamoto University, Department of Mathematics, Associate Professor, 理学部, 助教授 (30208665)
KOHNO Mitsuhiko  Kumamoto University, Department of Mathematics, Professor, 理学部, 教授 (30027370)
YAMAKI Hiroyoshi  Kumamoto University, Department of Mathematics, Professor, 理学部, 教授 (60028199)
岡 幸正  熊本大学, 理学部, 助教授 (50089140)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 1998: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1997: ¥2,300,000 (Direct Cost: ¥2,300,000)
Keywordsgeneral hypergeometric function / de Rham theory / Gauss hypergeometric / homology / cohomology / flat basis / intersection theory / quadratic relation / ガンマ関数 / 二次関形式 / 交点数 / hypergeometric / Airy / Okubo system / de Rham cohomology
Research Abstract

The objecitve of this project is to study the general hypergeometric functions (GHF) which were introduced by us to give a unified understanding of the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function and to give a natural generalization to the case of several variables.
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 whose integrand is a multivalued function on P^r. To obtain explicit resutis on GHF, it is important to understand this integral representation in the framework of de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here, for the integral on P^r, we defined the homology group as a locally finite homology group and then show that it is isomorphic to the relative homology group with compact supports for some pair of subsets P^r. Moreover … More , using this result, we computed explicitly, in the case r=1, the dimension of the homology group and gave a basis of the group.
2 : For the Beta function B(alpha, beta), the simplest case of GHF with regular singularity, and for the Gamma function GAMMA(alpha), the simplest case of GHF with irregular singularity, the following formulas are well known :
B(alpha, beta)B(-alpha, -beta)=2pii(<@D71(/)alpha@>D7+<@D71(/)beta@>D7)(<@D7-e<@D12pii(alpha+beta)@>D1-1(/)e<@D12piialpha@>D1-1(e<@D12piibeta@>D1-1)@>D7), gamma(alpha)gamma(1-a)=<@D7pi(/)sinpialpha@>D7
We investigate the problem of understanding the above formulas from the viewpoint of de Rham theory. Explicitly we try to understand the right hand sides of the above formulas as a product of cohomological intersection number and the homological intersection number. For the GHF defined by the 1-dimensional integral, we computed explicitly the intersection matrix for the cohomoloy group by choosing its good basis.
By the choice of good basis, we can show that the intersection matrix turns out to be independent of the variables of the general hypergeometric function. The main reason for the computability of the intersection numbers is that the good basis has, at each singular point of the connection form of the de Rham complex, the analogous properties to the flat basis of the Jacobi ring for the simple singlarity of A-type. Less

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] H. Kimura: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I, P^r case." Royal Soci Ediuburg. 127. 145-155 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H. Kimura: "On rational de Rham cohomology associated with the generalized Airy function" Annali di Scuola Norm. Sup. di Pisa. 24. 351-366 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura: "On the homology associated with the general Airy integral" Kumamoto J. Math.10. 11-29 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura & M.Taneda: "Analogue of flat basis and cohomology intersection number for general hypergeometric functions" Journal of Math. Sciences, the Tokyo Univ.62. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Haraoka: "Quadrotic relations for confluent hypergeometric functions on Z_2.ntl" Funkc. ER vac.42. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Chigira, N.Iiyori & H.Yamaki: "Non-abelian Sylow subgroups of finite groufss of even order." ERA Amer. Math. Soc.4. 88-90 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 河野實彦: "微分方程式と数式処理" 森北出版株式会社, 313 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 河野實彦: "Global Analysis in Linear Differential Equations" Kluwer Academic Publishers, (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I,P^1 case" Royal Soc.Edinburg. 127. 145-155 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura: "On rational de Rham cohomology associated with the generalized Airy function" Annali di Scuola Norm.Sup.di Pisa. 24 (to appear). 351-366 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura: "On the homology group associated with the general Airy integral" Kumamoto J.Math.10. 11-29 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kimura and M.Taneda: "Analogue of flat basis and cohomological intersection number for general hypergeometric functions" Journal of Mathematical Sciences, the Tokyo University. 62. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Haraoka: "Confluence of cycles for hypergeometric functions on Z_<2, n+1>" Trans.Amer.Math.Soc.349. 675-712 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Haraoka: "Quadratic relations for confluent hypergeometric functions on Z_<2, n+1>" Funkc.Ekvac.42. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Chigira, N.Iiyori and H.Yamaki: "Non-abelian Sylow subgroups of finite groups of even order" ERA Amer.Math.Soc.4. 88-90 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Chen, N.Chigira and H.Yamaki: "Finite groups with metacyclic automorphism groups" Northeast.Math.J.14. 5-8 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Khono: Differential Equations and algebraic manipulations. Morikita-Shuppan (Japanese), 313 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Khono: Global Analysis in Linear Differential Equations. Kluwer Academic Publishers, (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Chigira,N.Iiyon & H.Yamaki: "Non-abelian Sylous subgroups of finite groups of even order" ERA Amer.Math.Soc.4. 88-90 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] G.Chen,N.Chigira & H.Yamaki: "Finite groups with metacyclic automorphism groups" Northeast. Math. J.14. 5-8 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Haraoka: "Quadratic relations for confluent hypergeometric functin on Z_<2,n+1>" Funkc.Equrac.42. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 河野 實彦: "微分方程式と数式処理" 森北出版株式会社, 313 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 河野 實彦: "Global Analysis in Linear Differential EQuations" Kluwer Academic Publishers, (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hironobu KIMURA: "On Rational de Rham cohomology associated with the Generalized Airy function" Annali della Scuora Norm.di Pisa. 24. 351-366 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Haraoka Yoshishige: "Monodromyof an Okubo System with Non-Semisimple Exponents" Funkcial.EKvac.40・3. 435-457 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yoshishige Haraoka: "Confluence of cycles for hypergeometric functions on Z_<2,n+1>" Transaction of the American Math,Society. 349,2. 675-712 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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