Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

Matrix generalizations of some known classical special functions are important both from the theoretical and applied point of view (see, for example, [1–10]). *ese 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. *e results given by many authors [12, 20–24] follow as special cases in this work. *e 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. *e 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. *roughout this paper, let I and 0 denote the identity matrix and null matrix in C, respectively. A matrix A in C is a positive stable matrix if Re(λ)> 0 for all λ ∈ σ(A) where σ(A) denotes the set of all eigenvalues of A. In [12], if f(z) and g(z) are holomorphic functions in an open set Ω of the complex plane and if A is a matrix in C for which σ(A) ⊂ Ω, then

In particular, various results of gamma, Euler's beta, and hypergeometric matrix functions have been presented and investigated (see, e.g., [11][12][13][14][15][16][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. e results given by many authors [12,[20][21][22][23][24] follow as special cases in this work. e 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. e 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. roughout this paper, let I and 0 denote the identity matrix and null matrix in C r×r , respectively. A matrix A in C r×r is a positive stable matrix if Re(λ) > 0 for all λ ∈ σ(A) where σ(A) denotes the set of all eigenvalues of A. In [12], if f(z) and g(z) are holomorphic functions in an open set Ω of the complex plane and if A is a matrix in C r×r for which σ(A) ⊂ Ω, then (1) Let A be a positive stable matrix in C r×r ; then, the gamma matrix function in [11,12] is defined by (2) For positive stable matrices A and B ∈ C r×r , the beta matrix function in [11,12] is defined by Also, if A, B, and A + B are positive stable matrices in C r×r and AB � BA, then (4) From [12], for a matrix A ∈ C r×r , the matrix version of Pochammer symbol is defined as Gauss hypergeometric matrix function 2 F 1 (A, B; C; z) is given in [12] as follows: where |z| < 1 and A, B, and C ∈ C r×r such that C + nI is invertible for all integer. n ≥ 0. Also, the confluent hypergeometric matrix function is defined by and it satisfies the following matrix differential equation: where A and B ∈ C r×r and B + nI is invertible for every integer n ≥ 0. Furthermore, we have where A, B, and B − A are positive stable matrices and AB � BA.
In the recent paper [14], for any arbitrary parameter p with Re(p) > 0, the matrix generalizations of gamma and Euler's beta functions are given as follows: for p � 0, which gives gamma and Euler's beta matrix functions given by (2) and (3), respectively: and respectively, where A, B, X, and Y are positive stable matrices in C r×r and p is any arbitrary parameter with Re(p) > 0. Also, these are matrix versions of gamma and beta functions [23]. e case of A � B 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 A, B, A * , B * , and C * ∈ C r×r satisfying conditions that A, B, B * , C * − B * , and C * are the positive stable matrix, and and p be a number with Re(p) > 0. en, the generalized Gauss hypergeometric matrix function (GGHMF) is defined in [14] by and the generalized confluent hypergeometric matrix function (GCHMF) in the form (15) and (16), we have 2 Journal of Mathematics At p � 0 in (15), it reduces and also, at p � 0 in (16), we get Now, let us give the following definition as a generalization of the functions in (15) and (16).
and C * are positive stable matrices, and and C * i + kI is an invertible matrix for k ∈ N, i � 1, . . . , (s − 1), and p be a number with Re(p) > 0. en, the generalization hypergeometric-type matrix function is defined by where r and s ∈ N.

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: For the generalized gamma matrix function given by (12), we have the following integral representation: where B − A is a positive stable matrix in C r×r and AB � BA.
Proof. Using (9) in (12), we have Taking v � ut and η � t in the above equation, we can write en, by (10), we complete the proof of the theorem. □

Journal of Mathematics 3
Theorem 2.
e following integral representation for generalized Euler's beta matrix function in (13) holds well: where B − A is a positive stable matrix in C r×r and AB � BA.
Proof. e proof of the theorem is completed similar to eorem 1.
where the matrices S, X, Y, A, and B ∈ C r×r , S + X, Y + S, and X + Y + 2S are positive stable matrices, and SX � XS and SY � YS.
Proof. It is enough to use (13), interchange the order of integration, take transformations v � p/t(1 − t) and η � t in (13), and then use (12) in the left-hand side of the above equation, respectively.
where AY � YA and BY � YB.
□ Remark 2. If we take A � B and p � 0 in eorem 4, we obtain where X, Y, and X + Y are positive stable matrices and XY � YX.
Theorem 5. Let X and Y be matrices in C r×r satisfying conditions that X is a positive stable, 0 < Re(λ) < 1 for λ ∈ σ(Y), and XY � YX. en, we have Proof. By (13), we get On the contrary, if we consider Taylor expansion of (1 − t) − Y at t � 0, we can write en, using (13), the theorem can be proved.
Proof. From (12), by using the Leibnitz rule, it is easily seen that the left-hand side of the above equation can be written as where Z � 1 F 1 (A; B; − t − (p/t)). On the one hand, due to (8), we have Proof. In view of (13), the left-hand side of above equation is equal to where Z � 1 F 1 (A; B; − (p/t(1 − t))). On the other hand, by using equation (8), we obtain Journal of Mathematics

Properties of the GGHMF and GCHMF
In this section, we give some of the main results of the GGHMF and GCHMF as follows.
Proof. If we take in (15) and use the following relation: then we arrive at the required result.
□ Corollary 1. If we apply the substitution t � sin 2 v in (42), we derive and applying the substitution t � u/1 + u in (42), we obtain 6 Journal of Mathematics where A * C * � C * A * . Theorem 9. e following integral form for GGHMF holds true: where B − A is a positive stable matrix in C r×r and AB � BA.
Proof. From (15) and eorem 2, we have the desired relation. (B * ; C * ; z) has the next representation: Proof. One can easily prove the theorem similar to eorem 7.
Theorem 11. For the GGHMF with |arg(1 − z)| < π, then the following transformation formula holds true: Proof. In (42), by writing (1 − t) instead of t and using the following equation we obtain and also, for p � 0, and it is satisfied which is given in [25].
Remark 7. In the case of A � B in Corollary 3, we have and also, if we get p � 0, we find Corollary 4. For the GGHMF, the following transformation formula holds true: where |arg(1 + z)| < π. Theorem 12. e GGHMF F (A,B) (A * , B * ; C * ; z; p) verifies the recurrence relation: Proof. By using relation (42), we can write the left-hand side of (59) in the following form:  (1 − t))). On the other hand, it follows from (8): which proves the theorem. □ Theorem 13. For the GCHMF 1 F (A,B) 1 (B * ; C * ; z; p), the following recurrence relation holds true: Proof. It is enough to make similar calculations as in eorem 11.
where A, B, B * , C * , P, Q, and P + Q are positive stable matrices and commutative in C r×r with (Re(p) ≥ 0, |x| < 1), and the beta matrix transform of f(z) is defined as follows [25]: where P and Q are positive stable matrices in C r×r .
Proof. Using relation (64) and applying (15) to the beta matrix transform of (63), we have By interchanging the order of integration and summation with (4), we obtain Journal of Mathematics 9 which, according to (15), yields our desired result (63). is completes the proof of eorem 13.
where the Laplace transform of f(z) is defined as follows [26]: