Research Article Investigation of Fractional Calculus for Extended Wright Hypergeometric Matrix Functions

Throughout this paper, we will present a new extension of the Wright hypergeometric matrix function by employing the extended Pochhammer matrix symbol. First, we present the extended hypergeometric matrix function and express certain integral equations and di ﬀ erential formulae concerning it. We also present the Mellin matrix transform of the extended Wright hypergeometric matrix function. After that, we present some fractional calculus ﬁ ndings for these expanded Wright hypergeometric matrix functions. Lastly, we present several theorems of the extended Wright hypergeometric matrix function in fractional Kinetic equations.


Introduction and Preliminaries
Special functions are significant in many disciplines of mathematics nowadays because specific forms of these functions have become vital tools in several sciences such as mathematical physics, probability theory, computer science, and engineering (see [1,2]).
Special matrix functions demonstrate their relevance in addressing several physics issues, and their applications in statistics, lie groups, and differential equations are developing and becoming an active area in recent projects. Independent research is being conducted on new extensions of special matrix functions such as the beta matrix function, gamma matrix function, and Gaussian hypergeometric matrix function.
In this paper, the null matrix and identity matrix in ℂ r×r will be denoted as O and I, respectively. If a matrix ζ ∈ ℂ r×r , then, the spectrum of ζ is the collection of all eigenvalues of ζ and is represented by σðζÞ. A matrix ζ ∈ ℂ r×r is a positive stable if Re ðνÞ > 0 for all ν ∈ σ (ζ).
The Wright hypergeometric matrix function is defined in [10] as follows: where τ ∈ R + and ζ, η and ν are positive stable matrix in ℂ r×r and ν satisfies the condition (4). If η and ν are positive stable matrices function in ℂ r×r and ν satisfies the condition (4) then the Wright Kummer hypergeometric matrix function is defined in [10] as follows: This article is organized into five sections. In Section 2, we will provide a new extension of the Wright hypergeo-metric matrix function and prove some theorems about integral and derivative formula of the extension of the Wright hypergeometric matrix function 2 R τ 1 ½ðζ, ρÞ, η, ν ; z. In Section 3, we state the Mellin matrix transform of the extended Wright hypergeometric matrix function.
In Section 4, we applied certain fractional calculus ideas to the extended Wright hypergeometric matrix function. Lastly, in Section 5, we discuss several applications of 2 R τ 1 ½ðζ, ρÞ, η, ν ; z in fractional kinetic equations.
Definition 1. Let ζ, η, ν, and ρ be positive stable matrices in ℂ r×r and ν satisfies the condition (4) then the extended Gauss hypergeometric matrix function is given by Definition 2. Let ζ, η, ν, and ρ are positive stable matrices in ℂ r×r and ν satisfies the condition (4) then the extended Wright hypergeometric matrix function is where τ ∈ ð0, ∞Þ.
Proof. From (2) and (3), we find that Now, we can write This complete the proof.
Theorem 5. Let ζ, η, ν, κ, and ρ be matrices in ℂ r×r such that νη = ην and ρ, ν, η, and ν + κ are positive stable. Then, for jαzj < 1, we have Proof. We observe that substituting t = zu, we find that this completes the proof. Theorem 6. Let ζ, η, ν, and ρ be positive stable matrices in ℂ r×r then each of the following integrals hold true: 3 Abstract and Applied Analysis Proof.
put s = t 1/τ , and using the definition of beta matrix function, we have This can easily be written as and this finishes the proof of (i) 1 ½ðζ, ρÞ, η ; ν − τI ; ð1 − tÞ τ zdt by using the definition of 2 R ðτÞ 1 ½ðζ, ρÞ, η ; ν ; z, we find that and this can easily be written as This completes the proof.
Theorem 7. Let ζ, η, ν, and ρ be positive stable matrices in ℂ r×r then the following derivative formula hold true Proof. From the definition of extended Wright hypergeometric matrix function, we have Abstract and Applied Analysis This completes the proof.
Theorem 8. Let ζ, η, ν, and ρ be positive stable matrices in ℂ r×r , then the following derivative formula hold true: Proof. By using Definition (2) This finishes the proof,

Mellin Matrix Transform
Definition 9. Let FðζÞ be a function defined on the set of all positive stable matrices contained in ℂ r×r , then the Mellin transform is defined as follows: Such that the integral in right hand side exists.
The following lemma will be a useful tool in next theorem.