2018 Fiscal Year Final Research Report
Hypergeometric functions and Painleve equations
| Project/Area Number |
16K05165
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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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| Research Field |
Basic analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Research Collaborator |
Ebisu Akihito
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | 超幾何関数 / 隣接関係式 / 超幾何連分数 / 漸近解析 / 離散鞍点法 / 誤差評価 / ガンマ乗積表示 / パンルヴェ方程式 |
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| Outline of Final Research Achievements |
For generalized hypergeometric functions 3F2(1) we developed a general theory of contiguous relations. We established the linear independence of contiguous functions, existence and uniquness of contiguous relations, and algorithms for calculating their coefficients, as well as their group symmetry. As an application we constructed an infinite number of 3F2(1) continued fractions and determined exactly the leading asymptotics of their truncation errors. To do so we developed a discrete analogue of saddle point method for obtaining the asymptotic behavior of hypergeometric series containing a large parameter. As for Painleve equations we summarized the results obtained so far and set up the direction in which the next study should take. We also obtained some conditions for hypergeometric functions to admit gamma product formulas.
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| Free Research Field |
微分方程式論
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| Academic Significance and Societal Importance of the Research Achievements |
超幾何関数やパンルヴェ方程式で定義される関数は、数学や数理物理学のさまざまな局面に現れる重要な特殊関数である。そこで、これらの関数に特徴的な性質を調べることや、関連する関数の計算手段を確立することは、数学や数理物理学にとって大変重要である。また、これらの目的を達成するために、漸化式・差分方程式や漸近解析などの分野に、隣接関係式の同時性や離散鞍点法などの概念や手法を導入することは、解析学に新し知見をもたらすことになり、有益である。
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