A Class of Extended Hypergeometric Functions and Its Applications

TL;DR

This paper introduces a new class of extended hypergeometric functions with additional parameters, explores their fundamental properties, and applies them to derive inequalities and integral representations.

Contribution

It presents a novel extension of generalized hypergeometric functions and derives key properties and applications not previously available.

Findings

Derived transformation formulas and sum representations

Established Mellin-Barnes integral representations

Proved a Hardy-Hilbert type inequality involving extended hypergeometric functions

Abstract

Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and then put forward some fundamental results in the paper. Next, we will derive some properties of certain functions like extended Gauss hypergeometric functions, extended Appell's hypergeometric functions $\mathbf{F}_{1}^{(\kappa_{l})}$, $\mathbf{F}_{2}^{(\kappa_{l})}$, and extended Lauricella's hypergeometric function $\mathbf{F}_{D,(\kappa_{l})}^{(r)}$,$\mathbf{F}_{A,(\kappa_{l})}^{(r)}$, including transformation formulas, finite sum representations, Mellin-Barnes type integral representations and recurrence relations. Moreover, by using some new integral representation which will be presented in this paper, a Hardy-Hilbert type inequality involving extended Gauss hypergeometric functions will be established.