Asymptotic expansions for a class of generalized holomorphic Eisenstein series, Ramanujan's formula for $ζ(2k+1)$, Weierstrass' elliptic and allied functions (2201.10124v4)

Abstract: 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.