Alternating Mathieu Series, Hilbert–Eisenstein Series and Their Generalized Omega Functions

In this paper our aim is to generalize the complete Butzer–Flocke–Hauss (BFH) Ω-function in a natural way by using two approaches. Firstly, we introduce the generalized Omega function via alternating generalized Mathieu series by imposing Bessel function of the first kind of arbitrary order as the kernel function instead of the original cosine function in the integral definition of the Ω. We also study the following set of questions about generalized BFH Ω ν -function: (i) two different sets of bounding inequalities by certain bounds upon the kernel Bessel function; (ii) linear ordinary differential equation of which particular solution is the newly introduced Ω ν -function, and by virtue of the Čaplygin comparison theorem another set of bounding inequalities are given.In the second main part of this paper we introduce another extension of BFH Omega function as the counterpart of generalized BFH function in terms of the positive integer order Hilbert–Eisenstein (HE) series. In this study we realize by exposing basic analytical properties, recurrence identities and integral representation formulae of Hilbert–Eisenstein series. Series expansion of these generalized BFH functions is obtained in terms of Gaussian hypergeometric function and some bridges are derived between Hilbert–Eisenstein series and alternating generalized Mathieu series. Finally, we expose a Turán-type inequality for the HE series $$\mathfrak{h}_{r}(w)$$ .