Algebraic analysis of parametric Stokes phenomena

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400126
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kindai University
• Principal Investigator: AOKI TAKASHI 近畿大学, 理工学部, 教授 (80159285)
• Co-Investigator(Kenkyū-buntansha): 中村 弥生 近畿大学, 理工学部, 准教授 (60388494) ; 鈴木 貴雄 近畿大学, 理工学部, 准教授 (60527208)
• Co-Investigator(Renkei-kenkyūsha): HONDA Naofumi 北海道大学, 大学院・理学系研究院, 准教授 (00238817) ; KAWAI Takahiro 京都大学, 数理解析研究所, 名誉教授 (20027379) ; TAKEI Yoshitsugu 京都大学, 数理解析研究所, 准教授 (00212019) ; YAMAZAKI Susumu 日本大学, 理工学部, 教授 (00349953) ; KOIKE Tatsuya 神戸大学, 大学院・理学研究科, 准教授 (80324599) ; UMETA Yoko 城西大学, 理学部, 准教授 (90606386)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 超幾何微分方程式 / 超幾何関数 / WKB解 / 特異摂動 / ストークス現象 / 漸近展開 / ボレル総和法 / 合流型超幾何微分方程式

Outline of Final Research Achievements

Introducing a large parameter in the 3 parameters contained in the Gauss hypergeometric differential equation, we can construct the WKB solutions which are formal solutions to the equation. The construction is done algebraically and elementarily, however, these formal solutions are divergent in general and do not have analytic sense. We may apply the Borel resummation method to the formal solutions and can construct analytic solutions and bases of the solution space. On the other hand, the Gauss hypergeometric differential equation has standard bases of solutions expressed by the hypergeometric function. In this research, we have obtained linear relations between these two classes of bases. As an application, asymptotic expansion formulas with respect to the large parameter of the Gauss hypergeometric function have been obtained. At the same time, we have some formulas which describe the parametric Stokes phenomena of the WKB solutions.

Free Research Field

代数解析学