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

Let k be an arbitrary even integer, and E_k(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 E_k (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 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).