On Mathieu-type series for the unified Gaussian hypergeometric functions

Rakesh Parmar,Tibor Pogány

doi:10.2298/aadm190525014p

Abstract

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented.

Highlights

AND MOTIVATIONVarious extensions of Gauss’ hypergeometric function and other special functions were investigated recently by several authors, consult for instance [5]-[10], [15, 16], [19]–[22]
The importance of these functions is that they inherit most of the properties of the original functions and provide new relations between different special functions
We report here the following results [14, Corollary 2.7]

Summary

INTRODUCTION

Various extensions of Gauss’ hypergeometric function and other special functions were investigated recently by several authors, consult for instance [5]-[10], [15, 16], [19]–[22]. Generalized p–extended Beta function, Generalized p–extended Gauss’ hypergeometric function, integral representations, Mathieu–type series, Cahen formula, bounding inequality. Choi et al [9] considered the general Mathieu–type series and their alternating variants whose terms contain the (p, q)–extended Gaussian hypergeometric function Fp,q(z), and in turn, when p = q the p–extended Gaussian hypergeometric function Fp(z) and obtained the closed integral form expressions. We are interested in generalizing the integral expressions for the Mathieu–type series and its alternating variants whose terms contain the generalized p–extended Gauss’ hypergeometric function Fp(α,β;κ,μ)(z) which extends the results of the so–called p–extended Gaussian hypergeometric function Fp(z) recently developed by Choi et al [9].

BOUNDING INEQUALITIES FOR THE P –GENERALIZED MATHIEU–TYPE SERIES

DISCUSSION