Extension of Extended Beta, Hypergeometric and Confluent Hypergeometric Functions

Abstract

Abstract. Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

A New Extension of Beta and Hypergeometric Functions

“…Also, they defined further extension of extended hypergeometric and confluent hypergeometric functions with the help of newly defined beta function and established various properties, integral representations and differentiation formulas of extended hypergeometric and confluent hypergeometric functions. The authors conclude that if we letting λ = 1 throughout in the paper then all the results will be reduced to the work of Chaudhry et al (see [6]). Also, if we letting λ = 1 and p = q throughout in the paper then all the results will be reduced to results of extended beta function, extended beta distribution, extended Gauss hypergeometric and extended confluent hypergeometric functions ( see [4,5]).…”

“…It is clear that the beta distribution (4.1) is the extension of beta distribution defined by Chaudhary et al and Choi et al (see [4,6]). …”

“…One application of this newly extension of beta function B λ p,q (x, y) is to define beta distribution to a variables δ 1 and δ 2 with an infinite range, which is an extension of extended beta distribution defined by Chaudhary et al and Choi et al (see [4,6]). We define the extension of beta distribution by…”

“…The extended hypergeometric and confluent hypergeometric functions introduced in [6] by using the definition of extended beta function B p,q (δ 1 , δ 2 ) as follows: 15) where p, q ≥ 0 and ℜ(δ 3 ) > ℜ(δ 2 ) > 0, |z| < 1 and 16) where p, q ≥ 0 and…”

A new Generalization of Extended Beta and Hypergeometric Functions

“…The latest development and properties of such extension is found in the recent work of various researchers (see e.g., [1,2,3,4,7,9,12,11,13,14]). …”