Abstract
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 differential formulae concerning it. We also present the Mellin matrix transform of the extended Wright hypergeometric matrix function. After that, we present some fractional calculus findings for these expanded Wright hypergeometric matrix functions. Lastly, we present several theorems of the extended Wright hypergeometric matrix function in fractional Kinetic equations.
1. 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 will be denoted as and , respectively. If a matrix , then, the spectrum of is the collection of all eigenvalues of and is represented by . A matrix is a positive stable if for all ().
If and are holomorphic functions defined on an open set and if is a matrix in such that then (see [3]). Additionally, if is a matrix in such that and , then, . If is a positive stable matrix in , then, the gamma matrix function is defined as follows (see [3–6]):
If and are positive stable matrices in , then, the beta matrix function is defined by (see [3–6])
Also, if , and are positive stable matrices in such that then (see [3, 4])
If is a matrix in such that
then, the version Pochhammer matrix symbol is defined by (see [3])
From [7], if , are positive stable matrices in and satisfies condition (4), then the extended Gamma matrix function is defined by and the new extended Pochhammer matrix symbol is given by
The new extended Pochhammer matrix symbol is satisfying the following property (see [7]):
The Gauss hypergeometric matrix function is defined as follows (see [3, 8, 9]):
where and , satisfies the condition (4) and .
The Wright hypergeometric matrix function is defined in [10] as follows: where and , and are positive stable matrix in and satisfies the condition (4).
If and are positive stable matrices function in and 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 hypergeometric matrix function and prove some theorems about integral and derivative formula of the extension of the Wright hypergeometric matrix function . 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 in fractional kinetic equations.
2. The Extended Wright Hypergeometric Matrix Function
In terms of the generalized Pochhammer matrix symbol , we introduce the extended Gauss hypergeometric matrix function and the extended Wright hypergeometric matrix function as follows.
Definition 1. Let , , , and be positive stable matrices in and satisfies the condition (4) then the extended Gauss hypergeometric matrix function is given by
Definition 2. Let , , , and are positive stable matrices in and satisfies the condition (4) then the extended Wright hypergeometric matrix function is where .
Remark 3. Several particular remarks of the extended Wright hypergeometric matrix function are mentioned below: (i)When in (13), we get the Wright hypergeometric matrix function defined in (10)(ii)If we put and in (13), we get the Gauss hypergeometric matrix function as in (9)(iii)If and (where , , and are in ) in (13) then we get the Gauss hypergeometric function (see [11])
2.1. Integral and Derivative Formula of
In this part, we will provide integral representation and derivative formula of the extended Wright hypergeometric matrix function.
Theorem 4. Let , , , and be matrices in such that and , and are positive stable, then for , we have where .
Proof. From (2) and (3), we find that Now, we can write This complete the proof.
Theorem 5. Let , , , , and be matrices in such that and , , , and are positive stable. Then, for , we have
Proof. We observe that substituting , we find that this completes the proof.
Theorem 6. Let , , , and be positive stable matrices in then each of the following integrals hold true:
(i)
(ii)
Proof. (i)let From the definition of we have put , and using the definition of beta matrix function, we have This can easily be written as and this finishes the proof of (i) (ii)Let by using the definition of , we find thatand this can easily be written as This completes the proof.
Theorem 7. Let , , , and be positive stable matrices in then the following derivative formula hold true
Proof. From the definition of extended Wright hypergeometric matrix function, we have This completes the proof.
Theorem 8. Let , , , and be positive stable matrices in , then the following derivative formula hold true:
Proof. By using Definition (2) and differentiating term by term under the sign of summation, we have This finishes the proof,
3. Mellin Matrix Transform
Definition 9. Let be a function defined on the set of all positive stable matrices contained in , 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.
Lemma 10. Let , , , and are positive stable matrices in , then
Proof. From (31), the Mellin transform of in is From Fubini theorem with a little calculation (see [12]), we get This completes the proof.
Theorem 11. Let , , , , and be positive stable matrices in and satisfies the condition (4), then
Proof. This finishes the proof.
4. Fractional Calculus of the Extended Wright Hypergeometric Matrix Function
In this part, we will prove certain theorems concerning the Riemann-Liouville integral of the Wright hypergeometric matrix function. The fractional integral and derivative of Riemann-Liouville of order and are defined as follows (see [13, 14]):
If is a positive stable matrix in , such that then the following relation holds true (see [10]):
Theorem 12. Let , and be positive stable matrices in , such that , then for each , we have
Proof. From (37), we find that This completes the proof.
Theorem 13. Let , , , and be positive stable matrices in and such that then for each , we have
Proof. From (38), we have From Theorem (12), we find that Applying (29), we get the required result in (42).
5. Applications in Fractional Kinetic Equations
In our time, the fractional kinetic equations have a great importance in deferent branches of applied science such as astrophysics, control system, dynamic system, and mathematical physics.
The standard fractional kinetic equation is defined by where is the rate of reaction, and is the Riemann-Liouville fractional integral operator defined in (38). Furthermore, Saxena and Kalla (see [15]) considered the following fractional kinetic equations:
The Laplace transform of the Riemann-Liouville fractional integral operator is (see [16]) where is the Laplace transform of
Theorem 14. Let , , , and be positive stable matrices in such that is invertible, satisfies the condition (4) and , then the solution of the generalized fractional kinetic matrix equation: is given by where and called the generalized Mittag-Leffler matrix function (see [17, 18]).
Proof. Applying the Laplace transform on the equation (48) and using (47), we get
now we can write
where
Taking the inverse Laplace transform, we get
This completes the proof.
Theorem 15. Let , , , and be positive stable matrices in such that is invertible, satisfies the condition (4), such that and , then the solution of the generalized fractional kinetic matrix equation: is given by
Proof. By using the same steps of proof in the previous theorem, we get the required.
6. Conclusions
The topic of derivative with fractional parameter has lately attracted the attention of academics. For example, Riemann-Liouville developed the concept of fractional order derivative. Later, Caputo and others adjusted this fractional derivative. Because of their physical features, fractional derivatives have been successfully used to mimic numerous real-world issues. Recently, a derivative based on the classical derivative with a fractional parameter was developed. The derivative has highly fascinating qualities; hence, in this work, we have attempted to present some conclusions concerning fractional calculus of these extended Wright hypergeometric matrix functions as well as certain theorems of the extended Wright hypergeometric matrix function in fractional kinetic equations. As future work, and from a numerical point of view, we aim to employ some of the derived formulas in this paper along with suitable spectral methods to treat numerically the differential equations with polynomial coefficients.
Data Availability
Data supporting this manuscript are available from Scopus, Web of Science, and Google Scholar.
Conflicts of Interest
The authors declare that they have no conflicts of interest.