Generalization of Beta functions in terms of Mittag-Leffler function

Abstract

A new generalization of extended beta functions by using generalized Mittag-Leffler functions is proposed. Important properties of the generalized beta function and the integral representations are investigated. The generalization of the hypergeometric and Confluent hypergeometric functions is also introduced. Some more properties of these functions such as integral representations, differentiation formulas, transformation and summation formulas, Mellin transformations are also established.

New extension of beta, Gauss and confluent hypergeometric functions

“…The new extension of the extended beta function B ζ, α 1 α 2 ; m 1 , m 2 (a 1 , a 2 ), Gauss hypergeometric function , 1 2 ; 1 , 2 ( 1 , 2 ; 3 ; ) and confluent hypergeometric function , 1 2 ; 1 , 2 ( 2 ; 3 ; ) were obtained and presented with their important properties. The extended beta, Gauss and confluent hypergeometric functions and their special cases proposed in [21,[33][34][35][36][37]] can be regained from the newly proposed functions. It is hoped that it will be useful in Science and Technology [38][39][40].…”

“…The extended Gauss hypergeometric function is defined as (20)2. Special CasesSome special cases of the new extended beta function are Cases 1: When then the new extended beta function reduces to the beta function[21]:…”

A New Range of Modified Gamma and Beta Functions

“…Recently, many generalizations, modifications, extensions and variants of gamma and beta functions [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] have been proposed.…”