Fractional operators for the Wright hypergeometric matrix functions

In this paper, we contribute to the results of Bakhet et al. (Integral Transforms Spec. Funct. 30:138–156, 2019) by applying fractional operators to the Wright hypergeometric matrix functions. We give matrix recurrence relations and integral formulas for the Wright hypergeometric matrix functions. We also clarify particular cases of the main results.


Introduction
The studies of one-variable hypergeometric functions are more than 200 years old. These functions appear in the works of Euler, Gauss, Riemann, and Kummer (see [2]). Their integral representations were studied by Barnes and Mellin, and their special properties by Schwarz and Goursat. The famous Gauss hypergeometric equation is ubiquitous in mathematical physics, engineering, and mathematical sciences as many well-known partial differential equations may be reduced to the Gauss equation via separation of variables (see, e.g., [3][4][5][6]). Especially, in the last two decades, several generalizations of the well-known special functions have been studied by many mathematicians. This fact has inspired many mathematicians for investigations of several generalizations of hypergeometric functions.
The outline of the paper is as follows. Section 2 gives some elementary definitions and notions of this work. In Sect. 3, we obtain the matrix functions R(w; A, B; ν; λ), w, ν, λ ∈ C, and R(w; A, B; -μ; λ), w, μ, λ ∈ C, for parameter matrices A and B by using fractional operators. We also discuss some properties of these functions. Matrix recurrence relations of the Wright hypergeometric matrix function 2 R (τ ) 1 (A, B; C; z), z ∈ C, for parameter matrices A, B, and C and some particular cases are given in Sect. 4. In Sect. 5, we establish some (presumably) new integral representations of the Wright hypergeometric matrix function 2 R (τ ) 1 (A, B; C; z). Finally, some concluding remarks are presented in Sect. 6.

Notations and definitions
Let C N×N denote the vector space containing all square matrices with N rows and N columns with complex number entries, and let R(z) and I(z) denote the real and imaginary parts of a complex number z, respectively. For any matrix A in C N×N , σ (A) is the spectrum of A, the set of all eigenvalues of A, and where α(A) is referred to as the spectral abscissa of A, and α(-A) = -β(A). The square matrix A is said to be positive stable if β(A) > 0. By I and 0 we denote the identity matrix and the null matrix in C N×N , respectively. If f (z) and g(z) are holomorphic functions of the complex variable z defined in an open subset Ω of the complex plane and A is a matrix in C N×N such that σ (A) ⊂ Ω, then from the properties of the matrix functional calculus (see [9,[15][16][17]) it follows that The reciprocal gamma function denoted by Γ -1 (z) = 1 Γ (z) is an entire function of the complex variable z. Then the image of Γ -1 (z) acting on P, denoted by Γ -1 (A), is a welldefined matrix (see [18]).
The gamma matrix function Γ (A) and the beta matrix function B(A, B) have been defined in [9] as follows: and By application of the matrix functional calculus, for A in C N×N (see [9]) the Pochhammer symbol or shifted factorial is defined by under the condition that A + nI is invertible for all integers n ≥ 0.
Let A and B be commuting matrices in C N×N such that the matrices A + nI, B + nI, and A + B + nI are invertible for every integer n ≥ 0. Then we have [9] Definition 2.1 (see [12]) Let p and q be finite positive integers. Then the generalized hypergeometric matrix function is given by the matrix power series If p = 2 and q = 1, then (7) reduces to the Gauss hypergeometric matrix function (see

Definition 2.2 ([1]) Let
A, B, and C be positive stable matrices in C N×N satisfying condition (5). Then the Wright hypergeometric matrix function is defined by where τ ∈ R + = (0, ∞).
The Riemann-Liouville fractional integral of order ν such that Re(ν) > 0 is defined as [16,19,20] and the fractional differential operator of order μ such that Re(μ) > 0 is defined as where n is the smallest integer such that n > Re(μ). 15,16]) Let A be a positive stable matrix in C N×N Then the Riemann-Liouville fractional integral of order ν such that Re(ν) > 0 can be written as The Laplace transform of the function f (ζ ) is defined as [21] L

Fractional operators and the Wright-type hypergeometric matrix function
Consider the matrix function where A and B are positive stable matrices in C N×N such that B + nI are invertible for all integers n ≥ 0.
Applying the fractional integral operator (11) of order ν to f (w), we get which we can easily write in the following form: Here we denote (16) as R(w; A, B; ν; λ), that is, Now, applying the fractional differential operator (12) of order μ to f (w) defined in (15), we get which yields We denote (18) as

Some properties of R(w; A, B; ν; λ) and R(w; A, B; -μ; λ)
In this subsection, we study some of the main properties of the matrix functions R(w; A, B; ν; λ) and R(w; A, B; -μ; λ) by the following theorems.

Theorem 3.1 Let A and B be positive stable matrices in C N×N with B + nI invertible for
all integers n ≥ 0, and let λ, ν, μ ∈ C be such that |λw| < 1 and Re(μ) < 1. Then Proof From (11) and the left-hand side of (20) we get Putting u = zw, it follows that This is the proof of (20). From (12) and the left-hand side of (21) we obtain which is (21). Also, from (11) and the left-hand side of (22) we get which, upon substituting u = zw, yields This is the proof of (22). From (12) and the left-hand side of (23) we get which leads to (24).
From Theorems 3.1 and 3.2 we can obtain the following particular cases: 1. Taking A = a ∈ C 1×1 and B = b ∈ C 1×1 in Theorem 3.1, we get the classical results for the generalized hypergeometric function [22]. 2. Choosing C = c ∈ C 1×1 in Theorem 3.2, we get the classical results for the generalized hypergeometric function [22].

Recurrence relation of 2 R (τ ) 1 (A, B; C; z)
In this section, we give a matrix recurrence relation of Wright hypergeometric matrix function in the following theorem.
Proof Applying the fundamental relation of the gamma matrix function in (2) to (9), we can write This is proves (44) in Theorem 5.1.
Later on, many authors (see. e.g., [24][25][26]) introduced several extensions of the generalized (Wright) hypergeometric function. Very recently, the Wright hypergeometric matrix functions and incomplete Wright Gauss hypergeometric matrix functions were introduced by Bakhet et al. [1]. Here, with the help of the well-known fractional operators, we have obtained the Wright-type hypergeometric matrix functions and its properties. Also, we presented some properties of the Wright hypergeometric matrix function such as matrix recurrence relations and integral representations. Further research on this topic is now under investigation and will be reported in forthcoming papers.