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GENERALIZED WRIGHT FUNCTION AND ITS PROPERTIES USING EXTENDED BETA FUNCTION NABIULLAH KHAN, TALHA USMAN AND MOHD AMAN

Abstract. In this paper, we introduce a new generalization of the Wright function by using an extended beta function and study some classical properties of this function. We establish several formulas involving integral transforms (e.g. Jacobi transform, Gegenbauer transform) and the generalized family of Wright function that does not seem to be reported in the literature even for the basic Wright function. Furthermore, we discuss other results including the recurrence relation, derivative formula, fractional derivative formula and also a partly bilateral and partly unilateral generating relation for the generalized Wright function.


Introduction and preliminaries
The special functions of mathematical physics are found to be very useful in finding the solutions of initial and boundary value problems associated with partial differential equations and fractional differential equations. Special functions have widespread applications in areas of engineering as well, and often new perspectives in special functions are motivated by such connections. Several special functions, called now, special functions of fractional calculus play a very interesting role in the solution of fractional order differential equations such as Mittag-Leffler function, Wright function, and its auxiliary functions and Fox H -function.
The Wright function was first introduced [21,22,23,24] during the study of the asymptotic theory of partitions of natural numbers. It plays an important role in the solution of linear partial differential equations and subsequently, many other applications for instance, in the Mikusiński's operational calculus ( [7], see also [8]) and in the theory of Hankel type integral transforms have been found. Recently, this function has appeared in the area related to the partial differential equation of fractional order. Considering the boundary value problems for the fractional diffusion-wave equation, that is, the linear partial integro-differential equation 350 NABIULLAH KHAN, TALHA USMAN AND MOHD AMAN obtained from the classical diffusion or wave equation by replacing the first or second-order time derivatives by a fractional derivative of order α with 0 < α < 2, it was found the corresponding functions can be represented in terms of the Wright function. In this sequel, we intend to introduce a new generalization of Wright function by using the known extended beta function.
The classical Jacobi polynomials P in terms of hypergeometric function are defined as where, 2 F 1 is the familiar Gauss hypergeometric function.
The Jacobi transform formula for a function f (z) is defined as on condition that the integral in (1.11) exists. Now, on using the definition (1.10), (1.11) can be written as Lately, Srivastava et al. [17] introduced an interesting Jacobi transform formula in terms of the Kampé de Fériet function (see, for details, [9, p.27 et. seq.]).
It holds curiosity that the Jacobi polynomials P (α,β) n (z) as their special case, becomes other polynomials being the Legendre polynomials P n (z), the Chebyshev polynomials T n (z) and U n (z) (first and second kind), the Gegenbauer polynomials C ν n (z). We have indeed and respectively. In view of above relationships, the Gegenbauer transform G ν f (z); n and the Legendre transform L f (z); n (see [17]) can be defined as: We also introduce the two Wright type auxillary functions, respectively as

Integral formulas involving generalized Wright function
On interchanging the order of integration and summation in the equation above, which is proven under the given conditions here, we get Using (1.4) in the precedent equation, we get the desired result, and in view of (1.19) and (1.20), one can consequently get Now, letting t = u 1+u and t = sin 2 θ in (2.1), we come up with two interesting integral formulas for the extended Wright function in (1.18), which are given in the respective Corollaries.
where, the coefficients ℧ k are given by  (2.17) where ℧ k are given by (2.16
The special case of (3.1) when and p = 0 becomes the classical Reimann-Liouville fractional derivative.