A Class of Extended Hypergeometric Functions and Its Applications
Luo Minjie

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 , , and extended Lauricella's hypergeometric function ,, 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…
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Taxonomy
TopicsMathematical Inequalities and Applications · Mathematical functions and polynomials · Iterative Methods for Nonlinear Equations
