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

Abstract

In this paper, we obtain some generating matrix functions and integral representations for the extended Gauss hypergeometric matrix function EGHMF and their special cases are also given. Furthermore, a specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.

1. 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., [1–4] and elsewhere). New extensions of some of the well-known special matrix functions such as gamma matrix 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. These 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 [11–16]), 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.

The main object of this paper is to investigate various properties for the extended Gauss hypergeometric matrix function EGHMF. The 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.

Throughout this paper, I and 0 will denote the identity matrix and null matrix in , respectively. For a matrix , its spectrum is denoted by . We say that if for all , a matrix A in is a positive stable matrix. In [9, 17–19], if and are holomorphic functions in an open set of the complex plane and if A is a matrix in for which , then .

Notation 1. For all A in andthen the Pochhammer symbol is defined by [1, 20]By inserting a regularization matrix factor 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 ; then, the generalized Gamma matrix function is defined byfor 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 , and are positive stable and commutative matrices in satisfying spectral condition (1); then, the extended beta matrix function is defined byHence,For , it obviously reduces to the beta matrix function in [21] byFor , we will denote byLater, Abdalla and Bakhet [8] used to extend the Gauss hypergeometric matrix function in the following form:This matrix power series is seen to converge when Also, for , it reduces to the usual Gauss hypergeometric matrix function in [22]:where , and C are the matrices in and C satisfying condition (1).

Remark 1. is the special case of the well-known generalized hypergeometric matrix power series defined by [5, 20]For commutative matrices , and for , in such thatSome integral forms of the extended Gauss hypergeometric matrix function proved in [8] are given bywhere and C, B, and are positive stable.
For , we have the following differential formula [8]:

Definition 3 (see [20, 23]). Let A and B be positive stable matrices in then, the Jacobi matrix polynomial (JMP) is defined by

2. 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. They 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, 25–27]). Here, we present some generating functions involving the following family of the extended Gauss hypergeometric matrix functions:

Theorem 1. Let be given in (8); then, the following generating function holds true:

Proof. For convenience, let the left-hand side of (16) be denoted by . Applying the series expression of (8) to , we obtainBy changing the order of summations in (17), we obtainFurthermore, upon using the generalized binomial expansion, we find that the inner sum in (18) yieldsFinally, in view of (18) and (19), we obtain the desired result of Theorem 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 as follows:where, for convenience, 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 . The 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.

Theorem 2. For each , the following generating function holds true:where , and

Proof. Using the definitions (20) and (8) and changing the order of summation, the left-hand side of the result (22) is given byNow, by appealing once again to (19), we easily arrive to the desired result (22) asserted by Theorem 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 and replace A by in (22), we readily obtain assertion (16) of Theorem 1(ii)At , we observe that result (16) corresponds with that in [28](iii)For arbitrary complex numbers putting , and in (16) and (22), we find generating functions for the generalized Gauss hypergeometric function on [13, 27]

3. 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 given in (4), we introduce some new generalized integral formulas for the EGHMF in this section.

Theorem 3. For , the extended Gauss hypergeometric matrix function satisfies the following integral relations:(i)(ii)(iii) where , and are the positive stable matrices in

Proof. (i)Replacing by its integral representation (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 , we will have which is the required result in .(ii)Direct calculations using (12) yield Since then we have Thus, we get the desired assertion (ii) of Theorem 3.(iii)By using (12), it follows that Thus,Now, we put , , and in (29); we obtainHence, we easily arrive to the desired result (iii) asserted by Theorem 3:which proves the assertion of Theorem 3.

Remark 4. It is worthy of note that the special cases can be obtained from formulas , , and of Theorem 3 as follows:(1)Taking we find some integral representations for the Gauss hypergeometric matrix function (GHMF) (cf. [10, 20, 22])(2)Furthermore, choosing and setting and in Theorem 3, we obtain some integral representations for Gauss hypergeometric function (cf. [14])

4. An Application of the Computation of m Derivatives of the Extended Jacobi Matrix Polynomial

The 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.

Definition 5. Let A, B, and be positive stable matrices in whose eigenvalues, z, all satisfy . For any positive integer n, the nth extended Jacobi matrix polynomial is

Theorem 4. Let A, B, and be positive stable matrices in For the derivatives of extended Jacobi matrix polynomial, we findwhere and

Proof. Using (14) and (32) with the parameters , , and , we getfrom (34) and multiplying by , we obtainNow making use of the extended of Jacobi matrix polynomial (32), we findBy using (36), we getFurther simplification yieldsThis completes the proof of Theorem 4.

Theorem 5. Let A, B, C, D, and are positive stable matrices in and suppose that

Then,

Proof. Substitution of (40) into the RHS of (39) and making use of the extended of Jacobi matrix polynomial (32), we findsinceand we haveExpanding (43) and collecting similar terms, we obtainThis completes the proof of Theorem 5.
Finally, for the definition of extended of Jacobi matrix polynomial, we consider some of the extended special matrix polynomial as follows.

Definition 6. Let A and be positive stable matrices in whose eigenvalues, z, all satisfy . For any positive integer n, the nth extended ultraspherical matrix polynomials are

4.1. Special Cases

Upon assigning particular values to the parameters and variables, we interestingly get a range of special cases for (32) and (45) as discussed below:(i)Setting in (45), we find where is called the extended Chebyshev matrix polynomial of the first kind.(ii)Putting in (45), we have where is called the extended Chebyshev matrix polynomial of the second kind.(iii)Furthermore, taking in (45), we get where is called the extended Legendre matrix polynomial.(iv)For in (32), we get of Jacobi matrix polynomial in (15).(v)Taking and putting { and } in (32), we get Jacobi polynomial as follows:

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research group program under grant (R.G.P.1/148/40). This work was also supported by the National Natural Science Foundation of China (11601525).