### Abstract

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.

Original language | English |
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Title of host publication | Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava |

Publisher | Springer New York |

Pages | 775-808 |

Number of pages | 34 |

Volume | 9781493902583 |

ISBN (Print) | 9781493902583, 1493902571, 9781493902576 |

DOIs | |

Publication status | Published - Nov 1 2014 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava*(Vol. 9781493902583, pp. 775-808). Springer New York. https://doi.org/10.1007/978-1-4939-0258-3_30

**Alternating Mathieu series, Hilbert-Eisenstein series and their generalized omega functions.** / Baricz, A.; Butzer, Paul L.; Pogány, Tibor K.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava.*vol. 9781493902583, Springer New York, pp. 775-808. https://doi.org/10.1007/978-1-4939-0258-3_30

}

TY - CHAP

T1 - Alternating Mathieu series, Hilbert-Eisenstein series and their generalized omega functions

AU - Baricz, A.

AU - Butzer, Paul L.

AU - Pogány, Tibor K.

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84929902974&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929902974&partnerID=8YFLogxK

U2 - 10.1007/978-1-4939-0258-3_30

DO - 10.1007/978-1-4939-0258-3_30

M3 - Chapter

AN - SCOPUS:84929902974

SN - 9781493902583

SN - 1493902571

SN - 9781493902576

VL - 9781493902583

SP - 775

EP - 808

BT - Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava

PB - Springer New York

ER -