#### Abstract

Hypergeometric functions of matrix arguments occur frequently in multivariate statistical analysis. In this paper, we define and study extended forms of Gauss and confluent hypergeometric functions of matrix arguments and show that they occur naturally in statistical distribution theory.

#### 1. Introduction

The classical beta function, denoted by , is defined (Luke [1]) by the integral Based on the beta function, the Gauss hypergeometric function, denoted by , and the confluent hypergeometric function, denoted by , for , are defined as (Luke [1]) Further, using the series expansions of , , and in (2) and (3), respectively, series representations of hypergeometric functions and , for , are obtained as respectively.

From the confluent hypergeometric function , the Whittaker function (Whittaker and Watson [2]) is defined as where . Most of the properties and integral representations of the Whittaker function can be proved from those of the confluent hypergeometric function.

In 1997, Chaudhry et al. [3] extended the classical beta function to the whole complex plane by introducing in the integrand of (1) the exponential factor with . Thus, the extended beta function is defined as where . If we take in (7), then for and we have . Further, replacing by in (7), one can see that . The rationale and justification for introducing this function are given in Chaudhry et al. [3] where several properties and a statistical application have also been studied. Miller [4] further studied this function and has given several additional results.

In 2004, Chaudhry et al. [5] presented definitions of the extended Gauss hypergeometric function and the extended confluent hypergeometric function, denoted by and , respectively. These functions were introduced by considering the extended beta function (7) instead of beta function (1) in the general term of series (4) and (5). They defined these functions as Using the integral representation of the extended beta function (7) in (8) and (9), the integral representations of extended hypergeometric functions, for and , are obtained as

Substituting in (8) or (10), we have ; that is, the classical Gauss hypergeometric function is a special case of the extended Gauss hypergeometric function. Similarly, by taking in (9) or (11), we have , which means that the classical confluent hypergeometric function is a special case of the extended confluent hypergeometric function. Chaudhry et al. [5] found that the extended hypergeometric functions are related to the extended beta, Bessel, and Whittaker functions and also gave several alternative integral representations.

The classical functions, such as gamma, beta, confluent hypergeometric, Gauss hypergeometric, Bessel, and Whittaker, have been generalized to the matrix case and their properties have been studied extensively. For example, see Butler and Wood [6–8], Herz [9], Constantine [10], James [11], Muirhead [12], and Gupta and Nagar [13]. Many distributions of random matrices and their functions such as determinant and trace and moments of test statistics can be expressed in terms of hypergeometric functions of matrix arguments. For some recent work, the reader is referred to Bekker et al. [14, 15], Bekker et al. [16], and Gupta and Nagar [17]. Recently, Nagar et al. [18] have defined and studied the extended beta function of matrix argument.

The extended Gauss hypergeometric function and extended confluent hypergeometric function have not been generalized to the matrix case and therefore the main objective of this work is to define these generalizations, give various integral representations, study their properties, and establish their relationships with other special functions of matrix argument.

This paper is divided into eight sections. Section 2 deals with some well-known definitions and results on matrix algebra, multivariate gamma function, multivariate beta function, and special functions. In Section 3, the extended Gauss hypergeometric function of matrix argument has been defined and its properties have been studied. Section 4 deals with the extended confluent hypergeometric function of matrix argument and Section 5 defines the extended Whittaker function of matrix argument. Section 6 is devoted to several integrals involving these newly defined functions. The results contained in this section show the relationship of these functions with some known special functions. Finally, Sections 7 and 8 give a number of matrix variate distributions.

#### 2. Some Known Definitions and Results

This section provides definitions and important properties of some classical special functions that are critical to the development of this work.

Replacing the confluent hypergeometric function that appears in (6) by its integral representation (3), we obtain the integral representation of the Whittaker function as

Another integral representation of is obtained by substituting in (12), to get Further, application of Kummer’s transformation, namely, in (6) yields

Let be an matrix of real or complex numbers. Then, denotes the transpose of ; ; ; determinant of means that is symmetric positive semidefinite; means that is symmetric positive definite, means that both and are symmetric positive definite, and denotes the unique positive definite square root of .

Several generalizations of Euler’s gamma function are available in the scientific literature. The multivariate gamma function which is frequently used in multivariate statistical analysis is defined by (Ingham [19] and Siegel [20]) where and the integration is carried out over symmetric positive definite matrices. By evaluating the above integral, it is easy to see that Let be an symmetric positive definite matrix and make the transformation , where is the positive definite square root of , with the Jacobian . Then, The above result also holds for complex symmetric with by analytic continuation. The multivariate generalization of the beta function is given by where and .

Siegel [20] established the identity which can be derived from (19) by using the matrix transformation with the Jacobian .

The type 3 Bessel function of Herz (Herz [9, p. 517, p. 506]), , of symmetric positive definite matrix argument is defined by

The Gauss hypergeometric function of symmetric matrix argument , denoted by , is defined by where , , and . The confluent hypergeometric function of symmetric matrix argument , denoted by , is defined by where and . If we make the transformation in (22) and (23) with the Jacobian , we obtain alternative integral representations for and as Further, substituting with the Jacobian in (24), we get another interesting integral form of as Putting in (22) and evaluating the resulting integral using (19), one obtains where . Putting in (26) and using (20), one can easily show that Transforming , (23) becomes A comparison of (23) and (29) leads to the well-known Kummer’s relation (Herz [9, Eq. 2.8, p. 488]): Further, by using the transformation , (23) can be written as

For properties and further results on these functions, the reader is referred to Constantine [10], James [11], Muirhead [12], and Gupta and Nagar [13]. The numerical computation of a hypergeometric function of matrix arguments is very difficult. However, some numerical methods are proposed in recent years; see Hashiguchi et al. [21] and Koev and Edelman [22].

Also, in 1968, Abdi [23] defined the Whittaker function of matrix argument expressing it in terms of a confluent hypergeometric function of matrix argument as where . He also studied several properties and integral representations of this function. It is apparent that, by using different integral representations of in (32), a variety of integral representations for can be obtained. For example, using (31) in (32), we get

Next, we give definition and properties of the extended beta function of matrix argument due to Nagar et al. [18].

*Definition 1. *The extended matrix variate beta function, denoted by , is defined as
where and are arbitrary complex numbers and .

From the definition, it is apparent that the function is invariant under the transformation , ; thereby, is a function of the eigenvalues of the matrix . If we take in (34), then for and we have . Further, replacing by in (34), one can show that .

Now, applying the transformation in (34) with the Jacobian , we arrive at If we take in (35) and compare the resulting expression with (21), we obtain an interesting relation between the extended matrix variate beta function and the type 3 Bessel function of Herz as Also, from (20) and (35), one can prove the inequality

Let be a scalar valued function of an symmetric positive definite matrix such that , . Then, the -transform of , denoted by , is defined by where .

The -transform of the extended beta function of the matrix argument is given by where , , and .

#### 3. Extended Gauss Hypergeometric Function of Matrix Argument

In this section, we define the extended Gauss hypergeometric function of matrix argument (EGHFMA), which is a matrix variate generalization of the extended Gauss hypergeometric function (10) and an extended form of the classical Gauss hypergeometric function of matrix argument defined in (22). We also give several integral representations and properties of this function.

*Definition 2. *The extended Gauss hypergeometric function of matrix argument (EGHFMA), denoted by , is defined for an symmetric matrix as
where , , , and .

If we take in (40), then EGHFMA reduces to a classical Gauss hypergeometric function of matrix argument (22); that is, . Also, if we consider in (40) and compare the resulting expression with representation (34), we find that the extended beta function of matrix argument and EGHFMA are connected by the expression Further, substituting in (41) and using (36), we obtain

Theorem 3. *For , , and , we have . That is, , , is a function of the eigenvalues of the matrix . Further, for , which indicates that , , is a function of the eigenvalues of the matrix .*

*Proof. *Substituting with and replacing by , , in (40), we arrive at
where the last line has been obtained by substituting with the Jacobian and using (40). This means that , , is a function of the eigenvalues of the matrix . Similarly, if in (40) we take with and is replaced by , , we obtain
which shows that , , is a function of the eigenvalues of the matrix .

The following theorem gives an extended form of the integral representation given in (24).

Theorem 4. *Let be an symmetric matrix such that , , , and . Then,
*

*Proof. *In the integral representation of EGHFMA given in (40), substituting with the Jacobian , we obtain the desired result.

Theorem 5. *If is an symmetric matrix such that , , , and , then
*

*Proof. *From the* Trace Inequality* given in Abadir and Magnus [24, p. 338], it follows that which implies that and
Now, using the above inequality in the integral given in (45), we get
where the last line has been obtained by using (24). If is an positive definite matrix, then it has been shown in Abadir and Magnus [24, p. 333] that . This inequality, for , yields
which gives the second part of the inequality.

If we take in (46) and then use (41) and (27) in the resulting expression, we obtain where .

The following theorem gives -transform of the extended matrix variate Gauss hypergeometric function .

Theorem 6. *If is an symmetric matrix such that , , , and , then
*

*Proof. *Replacing by its integral representation given in (40) and changing the order of integration, we get
Now, using (18), we arrive at
Finally, the last integral is replaced by the Gauss hypergeometric function of matrix argument by using representation (22).

Substitution of in (51) gives the following interesting relationship between EGHFMA and classical Gauss hypergeometric function of matrix argument: Also, if we take in (51) and then use (41) and (27) in the resulting expression, we obtain the -transform of the extended beta function of matrix argument as where with .

The transformation formula for the extended Gauss hypergeometric function of matrix argument is given next.

Theorem 7. *If is an symmetric matrix such that , , , and , then
*

*Proof. *Making the transformation in the integral representation given in (40), one obtains
Now, writing
in (57) and noting that , we have
Finally, evaluating the above integral by using (40), we get the desired result.

It is noteworthy that in (56) gives the well-known transformation formula

#### 4. Extended Confluent Hypergeometric Function of Matrix Argument

In this section, we define and study the extended confluent hypergeometric function of matrix argument (ECHFMA), which is a generalization to the matrix case of the extended confluent hypergeometric function .

*Definition 8. *The extended confluent hypergeometric function of an symmetric matrix argument (ECHFMA), denoted by , is defined as
where , , and .

If we take in (61), then ECHFMA becomes the confluent hypergeometric function of matrix argument; that is, . Also, if we put in (61) and compare the resulting expression with (34), we will arrive at the conclusion that the ECHFMA and extended beta function of matrix argument retain the relationship

Theorem 9. *If and , then , . That is, is a function of the eigenvalues of the matrix . Further, for , which indicates that , , is a function of the eigenvalues of the matrix .*

*Proof. *The proof is similar to the proof of Theorem 3.

Theorem 10. *Let and be symmetric matrices with . If and , then
*

*Proof. *In the integral representation of the ECHFMA given in (61), consider the substitution with the Jacobian .

Corollary 11. *Let and be symmetric matrices with . If and , then
*

*Proof. *The desired result is obtained by evaluating the integral in (63) by using (61).

For , expression (64) reduces to the well-known Kummer’s relation for the classical confluent hypergeometric function of matrix argument. Moreover, the previous corollary is the generalization to the matrix case of Kummer’s relation for the extended confluent hypergeometric function of scalar argument.

Theorem 12. *If and , then
**
where and are symmetric matrices with .*

*Proof. *In the integral of the ECHFMA given in (61), consider the transformation , whose Jacobian is .

If we take in (65), we arrive at representation (25) of the classical confluent hypergeometric function of matrix argument.

Theorem 13. *Let , , , and be symmetric matrices with and . If and , then
**
where .*

*Proof. *Transforming with the Jacobian in representation (61), we obtain the result.

If we consider and in the above theorem, then we have

Theorem 14. *Let and be symmetric matrices, . If and , then
*

*Proof. *The proof is similar to the proof of Theorem 5.

The -transform of the extended matrix variate confluent hypergeometric function is given next.

Theorem 15. *If is an symmetric matrix, , , and , then
*

*Proof. *Replacing by its equivalent integral representation given in (61) and changing the order of integration, the integral in (69) is rewritten as
where the last line has been obtained by using (18). Finally, evaluating (70) using the definition of the confluent hypergeometric function of matrix argument, we get the desired result.

By putting , in (69), we get an interesting relation:

#### 5. Extended Whittaker Function of Matrix Argument

This section gives the definition of the extended Whittaker function of matrix argument, which is a generalization of the Whittaker function of matrix argument given in (32). Several properties and integral representations of this function are also derived.

*Definition 16. *The extended Whittaker function of matrix argument (EWFMA), denoted by , is defined for an symmetric matrix as
where and .

If we consider in (72), then the extended Whittaker function of matrix argument reduces to the classical Whittaker function of matrix argument given in (32); that is, . Several properties of the extended Whittaker function of matrix argument are inherited from the ECHFMA, so, as a consequence of Theorem 9, we have which indicates that the function with depends on the matrix only through its eigenvalues. Similarly, The above equation means that with depends on the matrix only through its eigenvalues.

An integral representation for the extended Whittaker function of matrix argument is obtained by replacing in (72) the integral representation of ECHFMA given in (61). In fact, Likewise, substitution of (67) in (72) yields the representation Clearly, when we take in the above expression, we obtain the integral representation (33) of the classical Whittaker function of matrix argument.

Theorem 17. *For symmetric matrix ,
**
where and .*

*Proof. *Using transformation (64) in (72), we have
Substituting (72) in the previous expression gives the result.

Theorem 18. *If and , then
*

*Proof. *The result follows by using inequality (68) in (72).

The following theorem gives the -transform of the extended Whittaker function of matrix argument.

Theorem 19. *If is an symmetric matrix, , and , then
*

*Proof. *Writing in terms of using (72), one obtains
Now, calculating the above integral by using (69) and then substituting the resulting expression in terms of Whittaker function of matrix argument, we get the final result.

Substitution of in the above theorem yields an interesting relationship between and as

#### 6. Relationship between EGHFMA, ECHFMA, and EWFMA

In this section, we derive some results that are related to EGHFMA, ECHFMA, and EWFMA.

Theorem 20. *Let , , and be symmetric matrices such that , , , and . If , , and , then
*

*Proof. *Using the integral representation (61) and changing the order of integration, we have
Now, by virtue of (18), we have
Finally, we use (40) to achieve the final result.

Corollary 21. *Let and be symmetric matrices such that and . If , , and , then
*

*Proof. *Application of transformation (64) yields