  For a class of generalized holomorphic Eisenstein series, we establish
complete asymptotic expansions (Theorems~1~and~2), which 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
Weierstra{\ss}' elliptic and allied functions (Corollaries~4.7--4.9). Crucial
r{\^o}les in the proofs are played by certain Mellin-Barnes type integrals,
which are manipulated with several properties of confluent hypergeometric
functions.
