No 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 ∆ H,k (of weight k attached to the upperhalf 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).

No abstract

A zeta-function associated with Kummer's confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan's formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.

No abstract