Asymptotic expansions for a class of generalized holomorphic Eisenstein series, Ramanujan’s formula for $$\zeta (2k+1)$$ , Weierstraß’ elliptic and allied functions

Abstract

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems 1 and 2). These, together with the explicit expression of the latter remainder (Theorem 3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem 4 and Corollaries 4.1–4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary 4.6) as well as for Weierstraß’ elliptic and allied functions (Corollaries 4.7–4.9). Crucial roles in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties Kummer’s confluent hypergeometric functions.