Abstract
In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.
1. Introduction
Matrix generalizations of some known classical special functions are important both from the theoretical and applied point of view (see, for example, [1–10]). These new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, life testing, and telecommunications.
In particular, various results of gamma, Euler’s beta, and hypergeometric matrix functions have been presented and investigated (see, e.g., [11–17]). Motivated by investigations of the extended gamma, beta, and Gauss hypergeometric matrix functions given in [14–16, 18, 19], we aim to derive certain properties of matrix generalizations of gamma, Euler’s beta, Gauss, and confluent hypergeometric functions. The results given by many authors [12, 20–24] follow as special cases in this work.
The plan of this work is described below:
In Section 2, we propose to derive some integral representations and recurrence relations for generalizations of gamma and Euler’s beta matrix functions. The various properties for the generalizations of Gauss and confluent hypergeometric matrix functions are investigated in Section 3. Finally, we end up with the conclusion in Section 4.
Throughout this paper, let and denote the identity matrix and null matrix in , respectively. A matrix in is a positive stable matrix if for all where denotes the set of all eigenvalues of . In [12], if and are holomorphic functions in an open set of the complex plane and if is a matrix in for which , then
Let be a positive stable matrix in ; then, the gamma matrix function in [11, 12] is defined by
For positive stable matrices and , the beta matrix function in [11, 12] is defined by
Also, if , are positive stable matrices in and , then
From [12], for a matrix the matrix version of Pochammer symbol is defined as
The Gauss hypergeometric matrix function is given in [12] as follows:where and such that is invertible for all integer.
Also, the confluent hypergeometric matrix function is defined byand it satisfies the following matrix differential equation:where and is invertible for every integer . Furthermore, we havewhere are positive stable matrices and .
In the recent paper [14], for any arbitrary parameter with , the matrix generalizations of gamma and Euler’s beta functions are given as follows:for , which gives gamma and Euler’s beta matrix functions given by (2) and (3), respectively:andrespectively, where are positive stable matrices in and is any arbitrary parameter with . Also, these are matrix versions of gamma and beta functions [23]. The case of in (12) and (13) gives us generalizations of gamma and Euler’s beta matrix functions defined by (10) and (11), respectively.
Moreover, author in [14] defined the generalized Gauss and confluent hypergeometric matrix functions as follows:
Definition 1. Let , and satisfying conditions that are the positive stable matrix, andand be a number with . Then, the generalized Gauss hypergeometric matrix function (GGHMF) is defined in [14] byand the generalized confluent hypergeometric matrix function (GCHMF) in the form
Remark 1. For in (15) and (16), we haveAt in (15), it reducesand also, at in (16), we get
Now, let us give the following definition as a generalization of the functions in (15) and (16).
Definition 2. Let satisfying conditions that are positive stable matrices, andand is an invertible matrix for , and be a number with . Then, the generalization hypergeometric-type matrix function is defined bywhere .
2. Properties of Generalizations of Gamma and Beta Matrix Functions
In this section, we drive some properties of generalized gamma and Euler’s beta matrix functions, which are defined by (12) and (13), as follows:
Theorem 1. For the generalized gamma matrix function given by (12), we have the following integral representation:where is a positive stable matrix in and .
Proof. Using (9) in (12), we haveTaking in the above equation, we can writeThen, by (10), we complete the proof of the theorem.
Theorem 2. The following integral representation for generalized Euler’s beta matrix function in (13) holds well:where is a positive stable matrix in and .
Proof. The proof of the theorem is completed similar to Theorem 1.
Theorem 3. For generalized Euler’s beta matrix function , we have the next property:where the matrices , are positive stable matrices, and .
Proof. It is enough to use (13), interchange the order of integration, take transformations and in (13), and then use (12) in the left-hand side of the above equation, respectively.
Theorem 4. For generalized gamma matrix function , we getwhere .
Proof. It follows straightforwardly from (12).
Remark 2. If we take and in Theorem 4, we obtainwhere are positive stable matrices and .
Theorem 5. Let be matrices in satisfying conditions that is a positive stable, for , and . Then, we have
Proof. By (13), we getOn the contrary, if we consider Taylor expansion of at , we can writeThen, using (13), the theorem can be proved.
Remark 3. For in Theorem 5, we getand for in (32), we have
Theorem 6. The following recurrence relation for the generalized gamma matrix function holds well:
Proof. From (12), by using the Leibnitz rule, it is easily seen that the left-hand side of the above equation can be written aswhere . On the one hand, due to (8), we haveThus, the proof is completed.
Remark 4. For in Theorem 6, we get
Theorem 7. Generalized Euler’s beta matrix function verifies the next recurrence relation:
Proof. In view of (13), the left-hand side of above equation is equal towhere . On the other hand, by using equation (8), we obtainwhich yields the desired result of Theorem 7.
Remark 5. The case of in Theorem 7 gives
3. Properties of the GGHMF and GCHMF
In this section, we give some of the main results of the GGHMF and GCHMF as follows.
Theorem 8. For the GGHMF , the following integral form holds true:where .
Proof. If we takein (15) and use the following relation:then we arrive at the required result.
Corollary 1. If we apply the substitution in (42), we deriveand applying the substitution in (42), we obtainwhere .
Theorem 9. The following integral form for GGHMF holds true:where is a positive stable matrix in and .
Proof. From (15) and Theorem 2, we have the desired relation.
Theorem 10. The GCHMF has the next representation:
Proof. One can easily prove the theorem similar to Theorem 7.
Corollary 2. If we take in (48), we derive
Theorem 11. For the GGHMF with , then the following transformation formula holds true:
Proof. In (42), by writing instead of and using the following equationwe obtain
Remark 6. For in Theorem 11, one can easily obtainand also, for ,and it is satisfied which is given in [25].
Corollary 3. From Theorem 11, the GGHMF satisfies the transformation formula:where .
Remark 7. In the case of in Corollary 3, we haveand also, if we get , we find
Corollary 4. For the GGHMF, the following transformation formula holds true:where .
Theorem 12. The GGHMF verifies the recurrence relation:
Proof. By using relation (42), we can write the left-hand side of (59) in the following form:where . On the other hand, it follows from (8):which proves the theorem.
Theorem 13. For the GCHMF , the following recurrence relation holds true:
Proof. It is enough to make similar calculations as in Theorem 11.
Next, the integral transforms for the GGHMF are given as follows:
Theorem 14. The following beta matrix transform formula holds true:where are positive stable matrices and commutative in with , and the beta matrix transform of is defined as follows [25]:where and are positive stable matrices in .
Proof. Using relation (64) and applying (15) to the beta matrix transform of (63), we haveBy interchanging the order of integration and summation with (4), we obtainwhich, according to (15), yields our desired result (63). This completes the proof of Theorem 13.
Theorem 15. If , and , then the Laplace transform holds true:where the Laplace transform of is defined as follows [26]:
Proof. From definition (68) and (15), we find thatBy interchanging the order of integration and summation and using Laplace transform, we havewhich, upon using (21), yields our desired result (67).
Theorem 16. If , , and , then the following Whittaker transform formula hold true:
Proof. Starting with in L.H.S of (71), we obtainBy interchanging the order of integration and summation, we obtainUsing the following integral form (cf. [26]),and (73) becomes the following equation:and relation (15) evidently leads us to the required result.
4. Conclusion
This manuscript is a continuation of the recent papers [14–16, 18, 19]. In the current paper, we introduced new properties of extension of the gamma and beta matrix function. We also introduced new extensions of the Gauss hypergeometric matrix function and confluent hypergeometric matrix function. Then, we discussed certain properties of these extended matrix functions such as the integral representations, transformation formulae, recurrence relations, and integral transforms. In addition, some interesting special cases of our main results are archived.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The second authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant no. (R.G.P-2/53/42).