Study of hypergeometric functions on the Grassmannian, q-hypergeometric functions and nonlinear special functions
| Project/Area Number |
15K04903
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kumamoto University |
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Principal Investigator |
Kimura Hironobu 熊本大学, 大学院先端科学研究部(理), 教授 (40161575)
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| 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)
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| 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 |
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| 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.
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| 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．
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Report
(6 results)
Research Products
(16 results)
