Takumi Noda

DIFFERENTIAL ACTIONS ON THE ASYMPTOTIC EXPANSIONS OF NON-HOLOMORPHIC EISENSTEIN SERIES

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).

On generalized Lipschitz-type formulae and applications

The purpose of this paper is to present a certain Lipschitz‐type formula and its generalization via Mellin‐Barnes type integrals. We introduce a double Eisenstein series defined on the pair of the upper and lower half planes, and show their transformation properties by using the generalized Lipschitz‐type formulae.