Study of general hypergeometric functions and integrable systems coming from monodromy preserving deformation

2015 Fiscal Year Final Research Report

• Project/Area Number: 23540247
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kumamoto University
• Principal Investigator: KIMURA Hironobu 熊本大学, 自然科学研究科, 教授 (40161575)
• Co-Investigator(Renkei-kenkyūsha): HARAOKA Yoshishige 熊本大学, 自然科学研究科, 教授 (30208665) ; NOUMI Masatoshi 神戸大学, 理学系研究科, 教授 (80164672) ; IWASAKI Katsunori 北海道大学, 理学研究院, 教授 (00176538) ; SAKAI Hidetaka 東京大学, 数理科学研究科, 准教授 (50323465)
• Research Collaborator: NAGOYA Hajime
• Project Period (FY): 2011-04-28 – 2016-03-31
• Keywords: 特殊関数 / 可積分系 / Twistor theory / Radon transform / 超幾何関数

Outline of Final Research Achievements

Among special functions, which have good properties, we know the Guass hypergeometric function and Painleve functions which can be characterized by differential equations, integral representations, and contiguity relations. Our study is to generalize and describe them in a unified way. This viewpoint enables to understand why the good properties hold for these objects. The general hypergeometric systems (GHGS) and the general Schlesinger systems (GSS), which generalize Gauss hypergeometric equation and Painleve equations, respectively, are both defined on the Grassmannian manifold. We gave the explicit form of monodromy preserving deformation which gives GSS. We studied, by examining the results of Shah and Woodhouse, when GSS has solutions expressed by the solutions of GHGS and how these solutions can be expressed using solutions of GHGS. As a by-product, we found the relation between the theory of semi-classical orthogonal polynomials and the particular solutions of GSS.

Free Research Field

解析学