TY - JOUR

T1 - Differential actions on the asymptotic expansions of non-holomorphic Eisenstein series

AU - Katsurada, Masanori

AU - Noda, Takumi

N1 - Funding Information:
This research was supported in part by Grant-in-Aid for Scientific Research (No. 16540038), The Ministry of Education, Science, Sports, Culture of Japan. A portion of the present research was made during the first author’s academic stay at Mathematisches Institut, Westfälische Wilhelms-Universität Münster. He would like to express his sincere gratitude to Professor Christopher Deninger and the institution for their constant support and warm hospitality.

PY - 2009

Y1 - 2009

N2 - 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).

AB - 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).

KW - Asymptotic expansion

KW - Eigenfunction equation

KW - Mellin-Barnes integral

KW - Non-homorphic Eisenstein series

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U2 - 10.1142/S1793042109002559

DO - 10.1142/S1793042109002559

M3 - Article

AN - SCOPUS:70350060198

VL - 5

SP - 1061

EP - 1088

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 6

ER -