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 [Formula: see text] as well as with a novel approach to Hilbert–Eisenstein series [Formula: see text], introduced by Michael Hauss in 1995. The latter turns 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.

Extension of Mathieu series and alternating Mathieu series involving the Neumann function $$Y_\nu $$

“…Remark 3. 5 As we can see, there exist certain oscillations in the graphics for the relative errors R S,n (r ) (Fig. 2 (right)) and R S,n (r ) (Fig.…”

“…The notation Ω(z), z ∈ C, stands for the so-called complete Butzer-Flocke-Hauss (BHF) Omega function introduced in [4,Definition 7.1], [5] in the form…”

“…In turn, recognizing that the same integral consist both Ω μ,ν (x) and S μ,ν (r ) in (7.2) and (4. which shows that Ω is intimately connected with the Eta function. In [5] the authors discussed the relations of the Mathieu-, and the alternating Mathieu series and its generalized variants from one, and the Ω(z) function from other side by the Taylor expansion of the Hilbert-Eisenstein series h 1 (z) of the first order and the polygamma function ψ (r ) of order r (see also [2]). However, our recent considerations are developed in another direction, according to the series representation of the expanded complete BHF Ω μ,ν (x) in terms of the Dirichlet Eta function.…”

“…while the real argument complete BHF Ω function's integral form by Tomovski and Pogány reads[29,p. 10, Theorem 3.3] By extensions in the integrand of the Butzer-Flocke-Hauss Omega function which is intimately connected to the generalized Mathieu series (consult the extensive study by Butzer and Pogány[5]) we are faced with a new territory of ideas and series/integral conclusion upon the structure of these kinds of generalizations.…”

A retrospective on research visits of Paul Butzer’s Aachen research group to Eastern Europe and Tenerife

“…Returning to Tibor Pogány, whom I met again at the "Symposium on Fourier Analysis, Interpolation and Signal Processing", conducted by Maurice Dodson at York 1988 (see [1]), it was with him that I wrote three joint papers [33][34][35] concerning the Omega function I had introduced with S. Flocke and M. Hauss in "Euler functions E α (z) with complex α and applications" [36].…”