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

TL;DR

This paper introduces new results and a novel approach to classical Eisenstein series and Hilbert-Eisenstein series, including differentiability, integral representations, and connections to the Omega function, expanding understanding of these special functions.

Contribution

It provides a new perspective and explicit formulas for Hilbert-Eisenstein series, linking them to digamma functions and the Omega function, with independent proofs of key properties.

Findings

New differentiability properties of Eisenstein series

Explicit integral representations of Hilbert-Eisenstein series

Connections established with the Omega function

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.