A fresh approach to classical Eisenstein series and the newer Hilbert-Eisenstein series

Abstract

This paper is concerned with new results for the circular Eisenstein series $\varepsilon_r(z)$ as well as with a novel approach to Hilbert-Eisenstein series $\mathfrak h_r(z)$, introduced by Michael Hauss in 1995. The latter turn out to be the product of the hyperbolic sinh--function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer--Flocke--Hauss Omega function, one basis for the study of Hilbert-Eisenstein series, which have been the subject of several recent papers.