2022 Fiscal Year Final Research Report
Connection problems of hypergeometric functions from the view point of higher dimensional Erdelyi cycles and their intersection numbers
| Project/Area Number |
19K03517
|
|---|
| 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 | Osaka University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2019-04-01 – 2023-03-31
|
| Keywords | 複素解析的線形微方程式 / 超幾何函数 / 接続問題 / Erdelyiサイクル / 交叉数 / Appellの超幾何函数 / Lauricellaの超幾何函数 / ねじれサイクル |
|---|
| Outline of Final Research Achievements |
We found integral representations of Appell's $F_2, F_3$, Horn's $H_2$ and Olsson's $F_P$ functions, and determined some connection formulas among them. We constructed a connection relation associated with Lauricella's $E_D$ equations, and, as its application, we give an affirmative answer to the conjecture by Shimeno-Tamaoka about the Harish-Chandra expansion of the Heckman-Opdam hypergeometric function of type $A$. We solved a connection problem associated with Lauricella's $E_A$ equations explicitly. We solved the connection problem associated with Apell's $E_1$ equation almost in the final form.
|
| Free Research Field |
解析学基礎，多変数超幾何函数の解の大域的性質の研究
|
| Academic Significance and Societal Importance of the Research Achievements |
本研究では，常微分方程式や完全積分可能な偏微分方程式である超幾何微分方程式の解に対する接続問題を解くことを主題としているが，複素解析的線形微分方程式の解の大域的性質を明らかにするために，その解がみたす接続関係を決定せよという問いは最も基本的であり究極的である．しかし，いっぽうで，接続問題が解けている例は非常に少ない．今回得た結果は，解の大域的理論のさらなる発展の礎になるものと期待される．
|
