A new generalization of beta function with three parameters Mittag-Leffler function

TL;DR

This paper introduces a novel three-parameter generalized beta function based on the Mittag-Leffler function, extending hypergeometric functions and deriving their key properties for advanced mathematical analysis.

Contribution

It presents a new generalized beta function with three parameters using Mittag-Leffler functions, along with related hypergeometric generalizations and their properties.

Findings

Derived integral representations and Mellin transforms

Established differentiation and transformation formulas

Obtained recurrence relations and summation formulas

Abstract

The main object of this paper is to present a new generalized beta function which defined by three parametres Mittag-Leffler function. We also introduce new generalizations of hypergeometric and confluent hypergeometric functions with the help of new generalized beta function. Furthermore, we obtained various properties of these functions such as integral representations, Mellin transforms, differentiation formulas, transformation formulas, recurrence relations and summation formula.