Further Generalized Beta Function with Three Parameters Mittag- Leffler Function

Salem Saleh Barahmah

doi:10.34198/ejms.1119.4149

Abstract

The purpose of the present paper is to introduce a new extension of extended Beta function by product of two Mittag-Leffler functions. Further, we present certain results including summation formulas, integral representations and Mellin transform.

Highlights

The classical Beta function is defined in [11]:∫ B(x, y) = 1t x−1(1 − t)y−1dt, (Re(x) > 0, Re( y) > 0) (1.1) =Γ(x) Γ(y) Γ(x + y)Re(x) > 0, Re( y) > 0, (1.2)where Γ(x) denotes the classical Gamma function defined in [11]∫ Γ(x) = ∞ t x−1e−tdt, (Re(x) > 0). 0 (1.3)In 1903, Mittag-Leffler [6] introduced the function Eα (z) defined by Received: November 19, 2018; Accepted: December 3, 2018
The purpose of the present paper is to introduce a new extension of extended Beta function by product of two Mittag-Leffler functions
The following two extended Beta functions are introduced by Chaudhry et al [4] and Choi et al [5] respectively:

Summary

Introduction

We present certain results including summation formulas, integral representations and Mellin transform. Keywords and phrases: extended Beta function, Mittag-Leffler function, summation formulas, integral representations, Mellin transform. Many authors have introduced certain extensions of extended Gamma and Beta functions (1.3) and (1.1) (see [1, 2, 3, 4, 5, 8, 9, 10]). The following two extended Beta functions are introduced by Chaudhry et al [4] and Choi et al [5] respectively:

Objectives

Results

Conclusion