Abstract
The main aim of this article is to study a new generalizations of the Gauss hypergeometric matrix and confluent hypergeometric matrix functions by using twoparameter Mittag–Leffler matrix function. In particular, we investigate certain important properties of these extended matrix functions such as integral representations, differentiation formulas, beta matrix transform, and Laplace transform. Furthermore, we introduce an extension of the Jacobi matrix orthogonal polynomial by using our generalized Gauss hypergeometric matrix function, which is very important in scattering theory and inverse scattering theory.
1 Introduction and preliminaries
A wide range of special functions in applied sciences are defined via improper integrals or infinite series. During last decades, several special functions become essential tools for scientists and engineering due to their applications in mathematical physics, engineering, and Lie theory. This inspire the study of the extensions of the special functions. In last few years, many extensions of gamma function, beta function, and Gauss hypergeometric functions have been studied by many researchers [4,5,6].
During the last two decades, Mittag–Leffler function has come into prominence after about nine decades of its discovery by a Swedish mathematician G.M. Mittag–Leffler, due to its vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences and world of fractional calculus. Many researches [7,8,9, 10,11] are continuously working on Mittag–Leffler functions and their properties. Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag–Leffler function plays an analogous role in the solution of noninteger order differential equations. In fact, the exponential function itself is a very special form, one of an infinite set of these seemingly ubiquitous functions. Mittag–Leffler function is entire function and contains several wellknown special functions as particular cases. That is why Mittag–Leffler function is very convenient for used as kernel in any integral equations.
In 1998, Jódar and Cortés [12,13] have introduced special matrix functions. These functions are found in the solutions of the physical problems as well as applications of these functions also increased in the statistics, Lie group theory, and differential equations. In recent years, matrix extensions of some known classical special function have become important tool of research.
On the other hand, the theory of orthogonal polynomials and special functions is of intrinsic interest to many parts of mathematics. Moreover, it can be used to explain many physical and chemical phenomena. Recently, the techniques of scattering theory and inverse scattering theory have been used to study the properties of orthogonal polynomials on a segment of the real line [1] and to investigate the properties of matrix orthogonal polynomials [2].
As we know that two highly developed theories of mathematical physics are those of orthogonal polynomials and potential scattering. While much of the work on orthogonal polynomials predates that on scattering theory, the latter has been much more intensively investigated in the last 25 years. In 1974, Case and Kac [3] have introduced that the theory of orthogonal polynomials sheds considerable light on the inverse problem of scattering theory and showed that methods of scattering theory form a unified basis for obtaining the various properties of orthogonal polynomials.
As motivated by earlier facts, here, we introduce a new generalization of hypergeometric matrix functions via twoparameter Mittag–Leffler matrix function and study important properties of these functions. Finally, we present an extension of the Jacobi matrix orthogonal polynomial by using our generalized hypergeometric matrix functions.
To discuss our main results, we need definition and results of some special matrix functions.
Throughout the article, let
The gamma matrix function is defined as ref. [12]:
where
Also if
The pochammer matrix symbol defined as ref. [13]:
From the aforementioned definition of the Pochammer matrix symbol and Eq. (1.2), we can observe that
The beta matrix function is defined as ref. [12]:
where
Also if
In continuation, Jódar and Cortés [13] defined a Gauss hypergeometric matrix function as follows:
where Matrices
The series (1.7) converges absolutely for
The confluent hypergeometric matrix function(Kummer’s matrix function) defined as refs [14,15]
where matrices
In 2013, Çekim inspired by the matrix generalizations has generalized the Gauss hypergeometric and confluent hypergeometric matrix functions by using confluent hypergeometric matrix function.
The generalized Gauss hypergeometric matrix function defined as ref. [16]:
The generalized confluent hypergeometric matrix function defined as ref. [16]:
where
In the aforementioned extensions, Çekim has used confluent hypergeometric matrix function as regularizer to extend Gauss hypergeometric and confluent hypergeometric matrix functions. They have introduced confluent hypergeometric matrix as kernel in definitions of Gauss hypergeometric and confluent hypergeometric matrix functions and studied several properties of these extended matrix functions.
In 2018, Abdalla and Bakhet [17] have extended the Gauss hypergeometric and confluent hypergeometric matrix functions by using the exponential matrix function.
The extended Gauss hypergeometric matrix function is defined as ref. [17]:
The extended confluent hypergeometric matrix function defined as follows [17]:
where
In the aforementioned extensions, Abdalla and Bakhet have used exponential matrix function as regularizer to extend Gauss hypergeometric and confluent hypergeometric matrix functions. They have introduced exponential matrix function as kernel in definitions of Gauss hypergeometric and confluent hypergeometric matrix functions and investigated many properties of these extended matrix functions.
In the sequence, Verma and Dwivedi have defined a new extensions of Gauss hypergeometric and confluent hypergeometric matrix functions by using confluent hypergeometric matrix function.
The new extended Gauss hypergeometric matrix function is defined as follows [18]:
The new confluent hypergeometric matrix function is defined as follows [18]:
where
In the aforementioned extensions, Verma and Dwivedi have used confluent hypergeometric matrix function as regularizer to extend Gauss hypergeometric and confluent hypergeometric matrix functions. They have introduced confluent hypergeometric matrix as kernel in definitions of Gauss hypergeometric and confluent hypergeometric matrix functions and studied some properties of these extended matrix functions.
Very recently, Goyal et al. [19] introduced an extension of the beta matrix function using the Wiman matrix function, thus studying various properties and relationships of that function.
Let P, Q, and X are positive stable matrices in
where
The 2parameter Mittag–Leffler matrix function defined as in ref. [21]:
where
In refs [16,18], Çekim, and Verma and Dwivedi have used confluent hypergeometric matrix function to regularize the Gauss hypergeometric and confluent hypergeometric matrix functions and Abdalla and Bakhet [17] have used exponential matrix function to regularize the Gauss hypergeometric and confluent hypergeometric matrix functions. In the aforementioned studies, they highlighted the important aspects of special matrix functions and inspired many researchers to work in this field. Later, motivated by the certain interesting recent above work done by authors [16,17,18] on generalizations of matrix functions with different special matrix functions. Here, we introduce new extensions of Gauss hypergeometric and confluent hypergeometric matrix functions with Mittag–Leffler type matrix function.
2 Main results
In this section, we consider new extensions of Gauss hypergeometric and confluent hypergeometric matrix functions by using twoparameter Mittag–Leffler matrix function as kernel in integral representations of these functions, and also we discuss some important basic properties of these extended matrix functions.
Definition 2.1
Let X,
where
Definition 2.2
Let X,
where
Remark
(i) If we set
and
(ii) If we take
and
Theorem 2.3
For positive stable matrices
where
Proof
Using the following known relation from Eq. (1.15):
in (2.1), we have:
On changing the order of integration and summation, we obtain:
Since
the last expression becomes the searched result:
Hence, the proof of the Theorem 2.3 is completed.□
By assigning particular values to the variable
Corollary 2.4
For new generalization of Gauss hypergeometric matrix function have following integral form hold true:
Proof
If we apply the substitution
Corollary 2.5
For new generalization of Gauss hypergeometric matrix function have following integral form hold true:
Proof
If we apply the substitution
Corollary 2.6
For new generalization of Gauss hypergeometric matrix function have following integral form hold true:
Proof
If we apply the substitution
Corollary 2.7
For new generalization of Gauss hypergeometric matrix function have following integral form hold true:
Proof
If we apply the substitution
Corollary 2.8
For new generalization of Gauss hypergeometric matrix function have following integral form hold true:
Proof
If we apply the substitution
Theorem 2.9
For positive stable matrices
where
The proof of Theorem 2.9 is similar to Theorem 2.3 and hence omitted.
Corollary 2.10
For new generalization of confluent hypergeometric matrix function has the following integral form hold true:
Corollary 2.11
For new generalization of confluent hypergeometric matrix function have following integral form hold true:
Corollary 2.12
For new generalization of confluent hypergeometric matrix function have following integral form hold true:
Corollary 2.13
For new generalization of confluent hypergeometric matrix function have following integral form hold true:
Corollary 2.14
For new generalization of confluent hypergeometric matrix function have following integral form hold true:
Proofs of Corollaries 2.10–2.14 are similar to Corollaries 2.4–2.8, respectively, and hence omitted.
Theorem 2.15
Let X,
where
Proof
Let
that is,
Replacing
Using the property of Pochhammer symbol
By exploiting (2.1), we get our statement:
Corollary 2.16
If we consider
Theorem 2.17
Let X,
where
Corollary 2.18
If we consider
Theorem 2.19
The Pfaff’s transformations for
where
Proof
From (2.3), we deduce:
Let
Thus, we get our desired result (2.28):
Corollary 2.20
For
where
Proof
If we apply the substitution
Corollary 2.21
For
where
Proof
If we apply the substitution
Theorem 2.22
The Kummer transformations for
where
Proof
From (2.9), we deduce:
On substitute
□
3 Integral transforms
In this section, we have introduced Beta matrix transform and Laplace transform of
Theorem 3.1
The following Beta matrix transform formula holds true:
where,
Proof
Applying Beta matrix transform to the
On changing the order of integration and summation,
Using (1.5), (2.1), and some rearrangements, we get our desired result.
Theorem 3.2
The following Laplace transform formula holds true:
where
Proof
Applying Laplace transform to the
On changing the order of integration and summation,
Using Laplace transform definition [23] and some rearrangements, we get our desired result.
□
4 An extended Jacobi matrix orthogonal polynomial
In this section, we introduce the extended Jacobi matrix orthogonal polynomial by using the extended Gauss hypergeometric matrix function (2.1). The Jacobi matrix orthogonal polynomial plays very important role in scattering and approximation theory [24].
Definition
For any positive integer
where
5 Conclusion
We conclude our analysis by remarking that the results presented in this article are new and very potential for the extension of other special matrix functions. First, we have generalized the Gauss hypergeometric and confluent hypergeometric matrix functions, then we have studied several basic properties like integral representations, differentiation formulas, and transformations for these extended hypergeometric functions. We have also derived two special integral transforms, beta matrix transform, and Laplace transform of these extended matrix functions. Finally, we defined the extended Jacobi matrix orthogonal polynomial, which appear in the scattering theory and inverse scattering theory.
The results presented in this article find an interesting application in the evaluation of certain infinite integrals whose specialized forms arise frequently in a number of applied problems. We will also use our main results to further extensions of Appell matrix and Lauricella matrix functions as well as their application in the scattering theory and inverse scattering theory.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions. Praveen Agarwal is thankful to the NBHM (DAE) (project 02011/12/2020 NBHM (R.P)/RD II/7867) for providing necessary support and facility. Also, the article and its translation were prepared within the framework of the agreement between the Ministry of Science and High Education of the Russian Federation and the Peoples Friendship University of Russia No. 075152021603: Development of the new methodology and intellectual base for the newgeneration research of Indian philosophy in correlation with the main World Philosophical Traditions.

Funding information: This article has been supported by the RUDN University Strategic Academic Leadership Program.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
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