Extension of gamma, beta and hypergeometric functions

A new generalization of confluent hypergeometric function and whittaker function

“…(α,β) σ (x, y),Özergin et al [7] further extended Gauss hypergeometric function and confluent hypergeometric function by…”

“…On taking m = n, (21) reduces to the integral representation of the generalized extended confluent hypergeometric function defined by Parmar [15], which further for n = 1 gives the integral representation of the extended confluent hypergeometric function given byÖzergin et al [7]. Further, if we put α = β and m = n in (21) then we get the integral representation of generalized confluent hypergeometric function defined by Lee et al [3] and if we set α = β and m = n = 1 in (21) then we obtain the integral representation of extended confluent hypergeometric function defined by Chaudhry et al [13].…”

“…Very recently, Parmar [15] introduced and investigated some fundamental properties and characteristics of more generalized beta type function B When m = 1, (2) reduces to the well-known generalized beta type function defined byÖzergin et al [7]: dt (ℜ(σ) > 0), (4) which was introduced by Chaudhry et al [12] in 1997. Clearly, classical beta function B(x, y) is given by B(x, y) = B 0 (x, y) = B (α,β) 0 (x, y).…”

An extension of Caputo fractional derivative operator and its applications

“…In recent years, many authors considered the several extensions of well known special functions (see, for example, [2,3,9,12,13]; see also the very recent work [8,10]). In 1994, Chaudhry and Zubair [4], introduced the generalized representation of gamma function.…”

Some applications of a hypergeometric identity

“…Also, they appear as solutions of many important ordinary differential equations [7,11,20]. Hence, finding any property of them may be valuable [1,5,17]. The generalized hypergeometric function p F q a 1 ; a 2 ; .…”