Study of differntial equations with actions of groups and its applications

2017 Fiscal Year Final Research Report

• Project/Area Number: 25287017
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Josai University
• Principal Investigator: Oshima Toshio 城西大学, 理学部, 客員教授 (50011721)
• Co-Investigator(Kenkyū-buntansha): 坂井 秀隆 東京大学, 大学院数理科学研究科, 准教授 (50323465)
• Co-Investigator(Renkei-kenkyūsha): KOBAYASHI Toshiyuki 東京大学, 数理科学研究科, 教授 (80201490)
• Research Collaborator: HIROE Kazuki ; NAKAMURA Akane ; HARAOKA Yoshishige ; MANO Toshiyuki ; SEKIGUCHI Jiro ; MIMACHI Katsuhisa ; SASAKI Ryu
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 微分方程式 / 超幾何関数 / 常微分方程式 / 数式処理 / KZ方程式

Outline of Final Research Achievements

Theory of linear ordinary differential equation with polynomial coefficients has a long history. In the study of rigid local system N. Katz introduced and formulated a transformation by fractional derivatives as a middle convolution. After his work there happens a novel development in the theory. Regarding the positions of singular points of the equations as new variables, we obtain KZ equations with several variables including Appell's hypergeometric equations. Applying this transformation to these equations, we analyze fundamental problems for these equations. For example, we construct these equations, give integral representations of their solutions, obtain the conditions of their irreducibility and moreover give a structure of the solutions of the original ordinary differential equations.

Free Research Field

代数解析学