2009 Fiscal Year Final Research Report
Behaviours of non-holomorphic Eisenstein series and the theory of q-hypergeometric functions
| Project/Area Number |
19540049
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Keio University |
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Principal Investigator |
KATSURADA Masanori Keio University, 経済学部, 教授 (90224485)
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| Co-Investigator(Kenkyū-buntansha) |
NODA Takumi 日本大学, 工学部, 准教授 (10350034)
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| Project Period (FY) |
2007 – 2009
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| Keywords | Eisenstein級数 / 漸近展開 / q超幾何関数 |
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| Research Abstract |
Let $E_k(s;z)$ be the non-holomorphic Eisenstein series with an even weight $k$ attached to the modular group $SL_2({\mathbb{Z})$. One of the major achievements of the present project is to establish its complete asymptotic expansion as $\Im z\to+\infty$ ; this yields various useful results on $E_k(s;z)$, which include its asymptotic expansion as $z\to0$ through the sector $0<\arg z<\pi$, its Fourier series expansion, and further the direct proof that it becomes the eigenfunction of the non-Euclidean Laplacian of weight $k$.
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