  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.
