Differential actions on the asymptotic expansions of non-holomorphic Eisenstein series

Masanori Katsurada, Takumi Noda

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Let k be an arbitrary even integer, and Ek(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL2 (ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of Ek (s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E0 (s;z) (due to the first author [16]) to that of Ek (s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on Ek (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 ΔH,k (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for Ek (s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2).

Original languageEnglish
Pages (from-to)1061-1088
Number of pages28
JournalInternational Journal of Number Theory
Issue number6
Publication statusPublished - 2009


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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