2023 Fiscal Year Final Research Report
Exact WKB analysis of parametric Stokes phenomena
| Project/Area Number |
18K03385
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Kindai University |
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Principal Investigator |
Aoki Takashi 近畿大学, 理工学部, 名誉教授 (80159285)
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| 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)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 超幾何関数 / 一般化超幾何微分方程式 / ヴォロス係数 / 接続公式 / 漸近展開 / ストークス現象 / ボレル変換 / 無限階微分作用素 |
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| Outline of Final Research Achievements |
The relation between the Borel sum of WKB solutions to the Gauss hypergeometric differential equation and the hypergeometric function is obtained. This result contains the description of the parametric Stokes phenomena for the WKB solutions to the equation. As an application, we obtained the asymptotic expansion formulas for the hypergeometric function with respect to the large parameter. Computation of the Voros coefficients of the hypergeometric differential equation is the key for this research. The explicit forms of the Voros coefficients for the generalized hypergeometric differential equation are also obtained for the origin and for the infinity. These expressions are compatible with the degeneration diagrams of the generalized hypergeometric differential equations.
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| Free Research Field |
解析学
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| Academic Significance and Societal Importance of the Research Achievements |
完全WKB解析は三十余前に始まり，多くの成果を挙げつつ発展してきたが，最近になりさらに多くの分野との結びつきが明らかになり，注目を浴びつつある．本研究で得られた成果は，完全WKB解析と古典解析の橋渡しを与えるという学術的意義を持つ．完全WKB解析の強みは，WKB解のボレル和を用いた関数の大域解析であるが，実際の問題に応用する際には古典的な解析関数との関係を明らかにする必要が生じる．本研究で与えた結果は，超幾何関数について，この部分を明らかにしている．併せて得られた超幾何関数の漸近展開公式は，将来的に公式集に含まれ，多くの応用で重要な役割を果たすことを期待している．
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