A generalized beta function and associated probability density

Abstract

We introduce and establish some properties of a generalized form of the beta function. Corresponding generalized incomplete beta functions are also defined. Moreover, we define a new probability density function (pdf) involving this new generalized beta function. Some basic functions associated with the pdf, such as moment generating function, mean residue function, and hazard rate function are derived. Some special cases are mentioned.

A generalization of beta-type distribution involving ω-Lauricella function in several variables

“…When x 2 = 0, we obtain the result given by Ben Nakhi and Kalla [17]. As ω → 1, we arrive at the representation of B * in terms of hypergeometric functions [16, p. 315].…”

“…It is interesting to note that if we set n = 2 in (7), then we obtain the generalised beta function recently studied by Ben Nakhi and Kalla [9, p. 322], which itself is the generalisation of the generalised beta function due to Ben Nakhi and Kalla [17].…”

“…defined as Function (67) includes the result given by Ban Nakhi and Kalla [9, p. 330, equation (33)], which itself is a generalization of their result given in ref. [17].…”

On extended M−series

“…(v) The generalized ω-Gauss hypergeometric function [21]: If we take p = 2, q = 1, ξ = µ = 1, then we have…”