A study on certain properties of generalized special functions defined by Fox-Wright function

In this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.


Introduction
On the last quarter century, some generalizations of special functions, which frequently used in applied mathematics, have been studied by many scientists [1, 7-14, 17, 19-29, 31-33]. Chaudhry and Zubair [10] defined the extended gamma function in 1994 as where Re(p) > 0. Three years later, Chaudhry et al. [7] defined the extended beta function as where Re(p) > 0, Re(x) > 0, Re(y) > 0. It clearly seems that, for p = 0, Γ 0 (x) = Γ(x) and B 0 (x, y) = B(x, y), where Γ(x) and B(x, y) are the classical gamma and beta functions [6]. In 2004, Chaudhry et al. [8] used B p (x, y) to extend the Gauss and confluent hypergeometric functions as follows: where p ≥ 0, Re(c) > Re(b) > 0. In the same paper, the authors also gave the integral representations of (1) and (2) as where p > 0, p = 0 and |arg(1 − z)| < π < p, Re(c) > Re(b) > 0, and Here (a) n is the Pochhammer symbol which defined as with the assume (a) 0 ≡ 1. The Fox-Wright function is given in [18] as where z, β i , µ j ∈ C, α i , κ j ∈ R, i = 1 . . . ξ and j = 1 . . . η. The asymptotic behaviour of the above function was studied by Fox [15,16] and Wright [34][35][36] for the large values of z, considering the condition If these conditions are met, for any z ∈ C the series (3) is convergent. For κ, µ, z ∈ C, Re(κ) > −1, the classic Wright function [18] can obtained by choosing ξ = 0 and η = 1 in equation (3). Inspired by the aforementioned studies and motivated by the frequent use of Fox-Wright function in the theory of special functions, we defined two new functions as generalizations of gamma and beta functions.

Generalized functions and their properties
Throughout the study, we assume that x, y, z ∈ C, k, m, n ∈ N, α i , κ j ∈ R, β i , µ j , a, b, c, p ∈ C, Re(p) > 0, Re(x) > 0, Re(y) > 0, and Re(c) > Re(b) > 0. For the sake of shortness, we did not wrote these conditions for the rest of the article, unless otherwise stated.
Let us defined the new generalizations aŝ We called them as ξ Ψ η -gamma and ξ Ψ η -beta functions.
Our first theorem is about the current relationship of the two ξ Ψ η -gamma functions.
Theorem 1. The following equality holds true: Proof. Substituting t = u 2 in (4), we get In the above equality, taking u = r(cos θ ) and v = r(sin θ ) yieldŝ which completes the proof.
Theorem 2. The ξ Ψ η -beta function has the following integral representations: which gives the result.
Proof. It is done by induction. The first order derivative of (5) is as follows: Let us assume that the k-order derivative of (5) is From the first order derivative of (6), the k + 1-order derivative is found as follows: This gives the result.
Theorem 4. The following equality is provided for Re(s) > 0: Proof. If we apply Mellin transformation according to argument p in equation (5), we have Letting v = p t(1−t) in (7), we get Thus, we have which completes the proof.
Remark 1. By using the inverse Mellin transform, it is easy to see Theorem 5. The following equality holds true: Proof. Direct calculation yields which is the result.
Theorem 6. The following summation formula is provided for Re(y) < 1: Proof. From the definition of the ξ Ψ η -beta function, we obtain With the help of the following series expression This completes the proof.
Theorem 7. The following equality holds true: Proof. From the definition of the ξ Ψ η -beta function, we get With the help of the following series expression which gives the result.

Proof. (5) equality provides the following equation
The derivative off (t : y; p) according to the parameter t provides the following equation: where d dt H(1−t) = −δ (1−t) and δ represents the Dirac delta δ (1−t) = δ (t −1) = 0 for t = 1. The relationship between the derivative of a function and the Mellin transformation is as follows: From here, by arranging, we find that Finally if x replaced by x + 1 and y replaced by y + 1 we get (8).

ξ Ψ η -generalization of Gauss and confluent hypergeometric functions
We used the ξ Ψ η -beta function (5) to define the generalizations of Gauss and confluent hypergeometric functions as respectively. We call ΨF p (a, b; c; z) as ξ Ψ η -Gauss hypergeometric function and ΨΦ p (b; c; z) as ξ Ψ η -confluent hypergeometric function.
Theorem 10. The ξ Ψ η -confluent hypergeometric function has the following integral representations: In the following theorems, we obtained the derivative formulas of ξ Ψ η -Gauss and ξ Ψ η -confluent hypergeometric functions with the help of the following equations: (a) n+1 = a(a + 1) n .
Theorem 11. The following equality holds true: Proof. The derivative of the ΨF p (a, b; c; z) according to the argument z is as follows: Replacing n → n + 1, we get Thus, the general form of the above equation is This completes the proof.
Theorem 12. The following equality is provided for Re(b) > 2, Re(c) > Re(b + 2): , Differentiatingf a,b,c (t : z; p) with respect to t we obtain .