Analysis of hypergeometric equations using various transformations

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03341
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Josai University
• Principal Investigator: Oshima Toshio 城西大学, 数理・データサイエンスセンター, 副所長 (50011721)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 超幾何微分方程式 / 多変数超幾何関数 / middle convolution / 積分変換 / 接続公式 / 不確定特異点 / Stokes係数 / 数式処理

Outline of Final Research Achievements

I analyze hypergeometric equations which characterize hypergeometric functions as special functions by applying several transformations including middle convolutions. I define versal unfoldings of ordinary differential equations with unramified irregular singular points and give versal integral representations of their solutions and a transformation theorem of Stokes coefficients of them. If the equations are rigid, I prove that they are versally extended to hypergeometric systems with several variables regarding singular points as variables.
Introducing new integral transformations, I analyze a wide class of hypergeometric systems with many variables which are not necessarily rigid and give integral representations of their solutions, power series expressions of them and their connection formulas etc.

Free Research Field

代数解析学

Academic Significance and Societal Importance of the Research Achievements

多変数の場合も含む超幾何関数について，満たす微分方程式系，べき級数表示，積分表示，既約性，隣接関係式などについての一般的解析手法を与え，それを用いて具体的で基本的な超幾何関数の場合の結果を示しており，今後の超幾何関数の研究にとって基本的な指針となるであろう．
手法は構成的で数式処理によるコンピュータアルゴリズムで実現しており，多くの研究者が利用可能なようにライブラリの形にまとめ，マニュアルも含めて公開している．