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

TL;DR

This paper derives complete asymptotic expansions for generalized holomorphic Eisenstein series, leading to new variants of classical formulas for the Riemann zeta-function and modular relations for elliptic functions.

Contribution

It establishes new asymptotic expansions and explicit remainder expressions for generalized Eisenstein series, extending classical formulas of Euler and Ramanujan.

Findings

Derived complete asymptotic expansions for Eisenstein series.

Extended Ramanujan's formulas to new variants involving zeta-values.

Established modular relations for elliptic and related functions.

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.