On the Extended Hypergeometric Matrix Functions and Their Applications for the Derivatives of the Extended Jacobi Matrix Polynomial

Generalizations of the classical special functions to matrix setting have become important during the previous years. Special matrix functions appear in solutions for some physical problems. Applications of special matrix functions also grow and become active areas in the recent literature including statistics, Lie groups theory, and differential equations (see, e.g., [1–4] and elsewhere). New extensions of some of the wellknown specialmatrix functions such as gammamatrix function, beta matrix function, and Gauss hypergeometric matrix function have been extensively studied in recent papers [5–10]. Hypergeometric matrix functions are an interesting problem to study from a purely analytic point of view. )ese functions arise in the study ofmatrix-valued spherical functions and in the theory of matrix-valued orthogonal polynomials. Moreover, they appear in the practice of various fields of mathematics and engineering, so knowledge of them is necessary for applications of theories associated with these fields. In various areas of applications, generating functions and integral transformations for some families of hypergeometric functions is potentially useful (see [11–16]), especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by ordinary and partial differential equations. )e main object of this paper is to investigate various properties for the extended Gauss hypergeometric matrix function EGHMF. )e generating functions and integral formulas are derived for EGHMF. We also present some special cases of the main results of this work. A specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed. )roughout this paper, I and 0 will denote the identity matrix and null matrix in C, respectively. For a matrix A ∈ C, its spectrum is denoted by σ(A).We say that if Re(ξ) for all ξ ∈ σ(A), a matrix A in C is a positive stable matrix. In [9, 17–19], if f(z) and g(z) are holomorphic functions in an open set Λ of the complex plane and if A is a matrix in C for which σ(A) ⊂ Λ, then f(A)g(A) � g(A)f(A).


Introduction
Generalizations of the classical special functions to matrix setting have become important during the previous years. Special matrix functions appear in solutions for some physical problems. Applications of special matrix functions also grow and become active areas in the recent literature including statistics, Lie groups theory, and differential equations (see, e.g., [1][2][3][4] and elsewhere). New extensions of some of the wellknown special matrix functions such as gamma matrix function, beta matrix function, and Gauss hypergeometric matrix function have been extensively studied in recent papers [5][6][7][8][9][10].
Hypergeometric matrix functions are an interesting problem to study from a purely analytic point of view. ese functions arise in the study of matrix-valued spherical functions and in the theory of matrix-valued orthogonal polynomials.
Moreover, they appear in the practice of various fields of mathematics and engineering, so knowledge of them is necessary for applications of theories associated with these fields.
In various areas of applications, generating functions and integral transformations for some families of hypergeometric functions is potentially useful (see [11][12][13][14][15][16]), especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by ordinary and partial differential equations. e main object of this paper is to investigate various properties for the extended Gauss hypergeometric matrix function EGHMF. e generating functions and integral formulas are derived for EGHMF. We also present some special cases of the main results of this work. A specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed. roughout this paper, I and 0 will denote the identity matrix and null matrix in C r×r , respectively. For a matrix A ∈ C r×r , its spectrum is denoted by σ(A). We say that if Re(ξ) for all ξ ∈ σ(A), a matrix A in C r×r is a positive stable matrix. In [9,[17][18][19], if f(z) and g(z) are holomorphic functions in an open set Λ of the complex plane and if A is a matrix in C r×r for which σ(A) ⊂ Λ, then f(A)g(A) � g(A)f(A).

Notation 1. For all A in C r×r , and
A + nI, is invertible for all integers n, By inserting a regularization matrix factor e − B/t , B ∈ C r×r . Abul-Dahab and Bakhet [6] have introduced the following generalization of the gamma matrix function. Definition 1. Let A and B be positive stable matrices in C r×r ; then, the generalized Gamma matrix function Γ(A, B) is defined by for B � 0 reduces gamma matrix function in [21].
Also, Abdalla and Bakhet [7] considered the extension of Euler's beta matrix function in the following definition.
Definition 2. Suppose that A, B, and P are positive stable and commutative matrices in C r×r satisfying spectral condition (1); then, the extended beta matrix function B(A, B; P) is defined by Hence, For P � 0, it obviously reduces to the beta matrix function in [21] by For p, q ∈ Z + , we will denote Γ( Later, Abdalla and Bakhet [8] used B(A, B; P) to extend the Gauss hypergeometric matrix function in the following form: is matrix power series is seen to converge when |z| < 1. Also, for P � 0, it reduces to the usual Gauss hypergeometric matrix function 1 F 2 (A, B; C; z) in [22]: where A, B, and C are the matrices in C r×r and C satisfying condition (1).
is the special case of the wellknown generalized hypergeometric matrix power series q F p (A i ; B j ; z) defined by [5,20] For commutative matrices A i , 1 ≤ i ≤ p, and for B j , 1 ≤ j ≤ q, in C r×r such that B j + nI are invertible for all integers n ≥ 0.
Some integral forms of the extended Gauss hypergeometric matrix function proved in [8] are given by where CB � BC and C, B, and C − B are positive stable. For (EGHMF), we have the following differential formula [8]: Definition 3 (see [20,23]). Let A and B be positive stable matrices in C r×r ; then, the Jacobi matrix polynomial (JMP)

Generating Functions of the EGHMF
In several areas in applied mathematics and mathematical physics, generating functions play an important role in the investigation of various useful properties of the sequences which they generate. ey are used to find certain properties and formulas for numbers and polynomials in a wide variety of research subjects, indeed, in modern combinatorics. One can refer to the extensive work of Srivastava and Manocha [24] for a systematic introduction and several interesting and useful applications of the various methods of obtaining linear, bilinear, bilateral, or mixed multilateral generating functions for a fairly wide variety of sequences of special functions (and polynomials) in one, two, and more variables, among much abundant literature; in this regard, in fact, a remarkable large number of generating functions involving a variety of special functions have been developed by many authors (see, e.g., [13,[25][26][27]). Here, we present some generating functions involving the following family of the extended Gauss hypergeometric matrix functions: (8); then, the following generating function holds true: Proof. For convenience, let the left-hand side of (16) be denoted by T. Applying the series expression of (8) to T, we obtain By changing the order of summations in (17), we obtain Furthermore, upon using the generalized binomial expansion, we find that the inner sum in (18) Finally, in view of (18) and (19), we obtain the desired result of eorem 1.
A further generalization of the extended Gauss hypergeometric matrix functions (8) is given in the following definition. □ Definition 4. In terms of the extended Gauss hypergeometric matrix function given by (8), we define a sequence Ω n (z) n∈N 0 as follows: where, for convenience, Δ(λ; A) abbreviates the array of λ matrix parameters: Remark 2. In the extended Gauss hypergeometric matrix function occurring in definition (20), it is understood that the matrix parameter A of definition (8) has been replaced by a set of λ parameters which are abbreviated by Δ(λ; A + nI). e above definition (20) is motivated by the extensive investigation on this subject. Now, we prove the following result, which provides the generating functions for the extended Gauss hypergeometric matrix functions defined above.
Proof. Using the definitions (20) and (8) and changing the order of summation, the left-hand side Υ of the result (22) is given by Now, by appealing once again to (19), we easily arrive to the desired result (22) asserted by eorem 2. □ Remark 3. Furthermore, we note the following special cases of generating functions of the EGHMF as follows: (i) It may be noted that if we set λ � 1 and replace A by A − kI in (22), we readily obtain assertion (16) of eorem 1 (ii) At P � 0, we observe that result (16) corresponds with that in [28] (iii) For arbitrary complex numbers α, β, c and η , (16) and (22), we find generating functions for the generalized Gauss hypergeometric function on [13,27]

Integral Representations for the EGHMF
Integral formulas with such special matrix functions such as the beta matrix functions and the hypergeometric matrix functions are used in solving numerous applied problems. Hence, their demonstrated applications and several generalizations of integral transforms with hypergeometric matrix functions have been actively investigated. Here, by means of the extended beta matrix function B(A, B; P) given in (4), we introduce some new generalized integral formulas for the EGHMF in this section.

Theorem 3.
For α ∈ C, the extended Gauss hypergeometric matrix function satisfies the following integral relations:  (12) and changing the order of integration, we get Now, if we integrate with respect to z using the properties of beta matrix function and substitute u � tz, we will have which is the required result in (i). (ii) Direct calculations using (12) yield dt.

An Application of the Computation of m Derivatives of the Extended Jacobi Matrix Polynomial
e Jacobi matrix polynomial and their special cases play important roles in approximation theory and its applications [20]. In this section, the extended Jacobi matrix polynomial ispresented and prove the following theorems for the mth derivatives of extended Jacobi matrix polynomials. Using the definition of the extended Gauss hypergeometric matrix functions EGHMF to define the extended matrix Jacobi polynomial and their special cases.
Proof. Using (14) and (32) with the parameters A � − nI, B � A + B + (n + 1)I, and C � A + I, we get Proof. Substitution of (40) into the RHS of (39) and making use of the extended of Jacobi matrix polynomial (32), we find is completes the proof of eorem 5. Finally, for the definition of extended of Jacobi matrix polynomial, we consider some of the extended special matrix polynomial as follows.