Abstract
The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.
Highlights
Introduction and PreliminariesAcademic Editors: Alberto Cabada, Manuel Manas and Sitnik SergeyReceived: 21 September 2021Accepted: 15 November 2021Published: 18 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.The theory of special functions, especially extensions of beta function, gamma function, hypergeometric functions has been one of the fastest rising investigate a topic in mathematical science because scientific researchers feel it is important to study the behavior of special functions with extended domains
In integral representation of Gauss hypergeometric function and confluent hypergeometric function, we introduce Wiman’s function as a kernel
After substitution of beta function value in terms of gamma function, in the above definition, we have another representation of a new extended confluent hypergeometric function (41): (r )
Summary
Introduction and PreliminariesAcademic Editors: Alberto Cabada, Manuel Manas and Sitnik SergeyReceived: 21 September 2021Accepted: 15 November 2021Published: 18 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.The theory of special functions, especially extensions of beta function, gamma function, hypergeometric functions has been one of the fastest rising investigate a topic in mathematical science because scientific researchers feel it is important to study the behavior of special functions with extended domains. We discuss convexity and inequalities related to extended beta function given in (31). From definition of extended beta function (31), we get our desired result: (u)
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