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Toward a unified theory special functions of several variables

Research Project

Project/Area Number 08454033
Research Category

Grant-in-Aid for Scientific Research (B)

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) YAMADA Kotaro  Kumamoto University, College of Liberal Arts and Sciences, Associate Professor, 理学部, 助教授 (10221657)
IKEDA Kaoru  Kumamoto University, Department of Mathematics, Associate Professor, 理学部, 助教授 (40232178)
OKA Yukimasa  Kumamoto University, Department of Mathematics, Associate Professor, 理学部, 助教授 (50089140)
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)
櫃田 倍之  熊本大学, 理学部, 教授 (50024237)
Project Period (FY) 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 1996: ¥2,900,000 (Direct Cost: ¥2,900,000)
Keywordsspecial 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
Research Abstract

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

Report

(2 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 木村 弘信: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I,P^1 case." The Royal Soc.Edinburg. 127. 145-155 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 木村 弘信: "On rational de Rham cohomology associated with the generalized Airy function" Annali di Scuola Norm.Sup.di Pisa. (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 木村 弘信: "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(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 山田 光太郎: "Surfaces of constant mean curvature c in H^3(-c^2)with prescribed hyperbolic Gauss map," Mathematische Aunalen. 304. 203-224 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 八牧 宏美, 飯寄 信保: "A conjecture of Frobenius" Sugaku Exposition. 9. 69-85 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] H.Kimura, T.Koitabashi: "Normalizer of maximal abelian subgroups of GL (n) and general hypergeometric functions" Kumamoto J.Math. 9. 13-43 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] H.Kimura: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I,P^1case" Royal Soc.Edinburg. 127. (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 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)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 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
      1996 Final Research Report Summary
  • [Publications] H.Kimura, K.Takano: "On confluences of the general hypergeometric systems" Preprint of Nice University. No 454. (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kotaro Yamada: "Surfaces of constant mean curvature c in H^3(-c^2)with prescribed hyperbolic Gauss map" Mathematische Annalen. 304. 203-224 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Nobuo Iiyori, Hiroyoshi Yamaki: "A conjecture of Frobenius" Sugaku Exposition. 9. 69-85 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Hironobu Kimura: "On rational de Rham cohomology associated with the generalized confluent hypergeometric function I,Ρ^1 case" The Royal Society of Edinburgh.(1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] Hironobu Kimura: "On rational de Rham cohomology associated with the generalized Airy function" Analli della Ecole Normale Superiore di Pisa. 24. (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] Hironobu Kimura: "On the homology group associated with the generalized Airy function" Kumamoto Journal of Mathematics. 10. (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] Hironobu Kimura: "Normalizer of maximal abelian subgroups of GL(n) and general hypergeometric functions" Kumamoto Journal of Mathematics. 9. 13-43 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Hiroyoshi Yamaki: "A conjecture of Frobenius" Sugaku Exposition. 10・1. 69-85 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Kotaro Yamada: "Surfaces of constant mean curvature cin H^3 (-c^2) with prescribed hyperbolic Gauss map" Mathematische Annalen. 304. 203-224 (1996)

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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