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Study of hypergeometric functions on the Grassmannian, q-hypergeometric functions and nonlinear special functions

Research Project

Project/Area Number 15K04903
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionKumamoto University

Principal Investigator

Kimura Hironobu  熊本大学, 大学院先端科学研究部(理), 教授 (40161575)

Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords超幾何関数 / 行列積分 / holonomic系 / 量子Painleve系 / 超幾何函数 / 行列積分型超幾何関数 / Painleve方程式 / 準直交多項式 / 特殊関数 / 量子パンルベ方程式 / Mellin-Barnes積分表示 / グラスマン多様体上の超幾何函数 / Schlesinger系 / モノドロミー保存変形 / 一般Schlesinger系 / Twistor theory / 一般超幾何関数
Outline of Final Research Achievements

The purpose of this research is to study the special functions and the integralble systems, such as the general hypergeometric functions(HGF) on the Grassmannian, the HGFs defined by matrix integral, and the general Schlesinger system, from the unified point of view using Twistor theory, Random matrix. We obtained the following results. (1) We study the relations between a class of semi-classical orthogonal polynomials, related with the general HGF, and a class of the general Schlesinger systems using Twistor theory. (2)We considered a matrix integral version of the Gauss hypergeometric, Kummer’s confluent hypergeometric, Bessel, Hermite-Weber and Airy. We found the relation of them to the theory of semi-classical orthogonal polynomials and to quantum Painleve equations. We also presented a conjecture for the systems of partial differential equations characterizing the functions defined by matrix integrals and showed the conjecture is true for some particular cases.

Academic Significance and Societal Importance of the Research Achievements

本研究は,特殊関数に関連する様々な結果を,できるだけ単純な原理から統一的に理解しようとするものである.特殊関数論はともすれば様々な公式の集積となってしまいがちであり,特殊関数個々の性質が統一的な視点がなく調べられる傾向がなきにしもあらずである.これらの性質が成り立つ根拠を明確にし,統一的な視点を導入することにより,専門家以外にもアプローチしやすくなり,他の科学分野との関連の発見や知見の深化に寄与することになると思われる.

Report

(6 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (16 results)

All 2019 2018 2017 2016 2015

All Journal Article (9 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 9 results,  Open Access: 3 results) Presentation (7 results) (of which Int'l Joint Research: 7 results,  Invited: 7 results)

  • [Journal Article] Three-term relations for 3F2(1)2018

    • Author(s)
      Akihito Ebisu and Katsunori Iwasaki
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 463 Issue: 2 Pages: 593-610

    • DOI

      10.1016/j.jmaa.2018.03.034

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Relation of Semi-Classical Orthogonal Polynomials to General Schlesinger systems via Twistor Theory2017

    • Author(s)
      Hironobu Kimura
    • Journal Title

      "Analytic, Algebraic and Geometric Aspects of Differential Equations", Trends in Mathematics

      Volume: 0 Pages: 399-414

    • DOI

      10.1007/978-3-319-52842-7_12

    • ISBN
      9783319528410, 9783319528427
    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Connection problem for regular holonomic systems in several variables2017

    • Author(s)
      Yoshishige Haraoka
    • Journal Title

      "Analytic, Algebraic and Geometric Aspects of Differential Equations", Trends in Mathematics

      Volume: 0 Pages: 337-350

    • DOI

      10.1007/978-3-319-52842-7_8

    • ISBN
      9783319528410, 9783319528427
    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Evaluation of the $BC_n$ elliptic Selberg integral via the fundamental invariants2017

    • Author(s)
      M. Ito, M. Noumi
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 145 Issue: 2 Pages: 689-703

    • DOI

      10.1090/proc/13234

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Hypergeometric series with gamma product formula2017

    • Author(s)
      Katsunori Iwasaki
    • Journal Title

      Indagationes Mathematicae

      Volume: 28 Issue: 2 Pages: 463-493

    • DOI

      10.1016/j.indag.2016.12.001

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Symmetric differential polynomials2017

    • Author(s)
      Kousuke Iida and Yoshishige Haraoka
    • Journal Title

      Kumamoto Journal of Mathematics

      Volume: 30 Pages: 7-14

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Confluence of general Schlesinger systems and Twistor theory2016

    • Author(s)
      Hironobu Kimura and Damiran Tseveennamjil
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 46 Pages: 289-309

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A generalization of the Sears--Slater transformation and elliptic Lagrange interpolation of type $BC_n$2016

    • Author(s)
      M. Ito, M. Noumi
    • Journal Title

      Adv. in Math.

      Volume: 229 Pages: 361-380

    • DOI

      10.1016/j.aim.2016.05.016

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] SOLUTIONS IN TERMS OF INTEGRALS OF MULTIVALUED FUNCTIONS FOR THE CLASSICAL HYPERGEOMETRIC EQUATIONS AND THE HYPERGEOMETRIC SYSTEM ON THE CONFIGURATION SPACE2016

    • Author(s)
      K. Mimachi, M. Noumi
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: 70 Issue: 2 Pages: 315-342

    • DOI

      10.2206/kyushujm.70.315

    • NAID

      130005421174

    • ISSN
      1340-6116, 1883-2032
    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Some special functions of matrix integral type and quantum Painleve equations2019

    • Author(s)
      木村弘信
    • Organizer
      International Symposium on Orthogonal Polynomials, Special Functions and Applications
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Error estimates for hypergeometric continued fractions2018

    • Author(s)
      岩﨑 克則
    • Organizer
      アクセサリ・パラメータ研究会
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On hypergeometric functions of matrix arguments2017

    • Author(s)
      Hironobu Kimura
    • Organizer
      Formal and Analytic Solutions of Diff. Equations (FASdiff17)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Middle convolution of monodromy for KZ type equation2017

    • Author(s)
      Yoshishige Haraoka
    • Organizer
      Formal and Analytic Solutions of Diff. Equations (FASdiff17)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Elliptic difference Painleve equation and elliptic hypergeometric integrals2017

    • Author(s)
      Masatoshi Noumi
    • Organizer
      XXVth International Conference on Integrable Systems and Quantum Symmetries
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Elliptic hypergeometric integrals and elliptic interpolation functions2016

    • Author(s)
      Masatoshi Noumi
    • Organizer
      BIRS workshop: ``Painelev\'e Equations and Discrete Dynamics''
    • Place of Presentation
      Banff International Reseach Station, Banff, Canada
    • Year and Date
      2016-10-03
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Orthogonal polynomials and General Schlesinger systems2015

    • Author(s)
      Hironobu Kimura
    • Organizer
      Analytic, Algebraic and Geometric Aspects of Differential Equations
    • Place of Presentation
      Poland, Bedlewo,
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2015-04-16   Modified: 2021-02-19  

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