### Abstract

Let k be an arbitrary even integer, and E_{k}(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL_{2} (ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of E_{k} (s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E_{0} (s;z) (due to the first author [16]) to that of E_{k} (s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on E_{k} (s;z), including its functional properties (Corollaries 2.1-2.3), its relevant specific values (Corollaries 2.4-2.7), and its asymptotic aspects as z → 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian Δ_{H,k} (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for E_{k} (s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2).

Original language | English |
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Pages (from-to) | 1061-1088 |

Number of pages | 28 |

Journal | International Journal of Number Theory |

Volume | 5 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2009 Oct 22 |

### Keywords

- Asymptotic expansion
- Eigenfunction equation
- Mellin-Barnes integral
- Non-homorphic Eisenstein series

### ASJC Scopus subject areas

- Algebra and Number Theory