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

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04903
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto 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
• Keywords: 超幾何関数 / 行列積分 / holonomic系 / 量子Painleve系

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.

Free Research Field

解析学

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．