Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications

In this article, we define and study generalized forms of extended matrix variate gamma and beta functions. By using a number of results from matrix algebra, special functions of matrix arguments and zonal polynomials we derive a number of properties of these newly defined functions. We also give some applications of these functions to statistical distribution theory.


Introduction
The gamma function was first introduced by Leonard Euler in 1729, as the limit of a discrete expression and later as an absolutely convergent improper integral, namely, Γ(a) = ∞ 0 t a−1 exp(−t) dt, Re(a) > 0. ( The gamma function has many beautiful properties and has been used in almost all the branches of science and engineering.
One year later, Euler introduced the beta function defined for a pair of complex numbers a and b with positive real parts, through the integral ( The In statistical distribution theory, gamma and beta functions have been used extensively.Using integrands of gamma and beta functions, the gamma and beta density functions are usually defined. Recently, the domains of gamma and beta functions have been extended to the whole complex plane by introducing in the integrands of ( 1) and ( 2), the factors exp (−σ/t) and exp [−σ/t(1 − t)], respectively, where Re(σ) > 0. The functions so defined have been named extended gamma and extended beta functions.
In 1994, Chaudhry and Zubair [7] defined the extended gamma function, Γ(a; σ), as where Re(σ) > 0 and a is any complex number.For Re(a) > 0 and σ = 0, it is clear that the above extension of the gamma function reduces to the classical gamma function, Γ(a, 0) = Γ(a).The extended gamma function is a special case of Krätzel function defined in 1975 by Krätzel [16].The generalized gamma function (extended) has been proved very useful in various problems in engineering and physics, see for example, Chaudhry and Zubair [8].
In 1997, Chaudhry et al. [6] defined the extended beta function where Re(σ) > 0 and parameters a and b are arbitrary complex numbers.When σ = 0, it is clear that for Re(a) > 0 and Re(b) > 0, the extended beta function reduces to the classical beta function B(a, b).
Recently, Özergin, Özarslan and Altın [24] have further generalized the extended gamma and extended beta functions as where Φ (α; β; •) is the type 1 confluent hypergeometric function.The gamma function, the extended gamma function, the beta function, the extended beta function, the gamma distribution, the beta distribution and the extended beta distribution have been generalized to the matrix case in various ways.These generalizations and some of their properties can be found in Olkin [23], Gupta and Nagar [10], Muirhead [18], Nagar, Gupta, and Sánchez [19], Nagar, Roldán-Correa and Gupta [20], Nagar and Roldán-Correa [21], and Nagar, Morán-Vásquez and Gupta [22].For some recent advances the reader is refereed to Hassairi and Regaig [12], Farah and Hassairi [4], Gupta and Nagar [11], and Zine [25].However, generalizations of the extended gamma and extended beta functions defined by ( 5) and (6), respectively, to the matrix case have not been defined and studied.It is, therefore, of great interest to define generalizations of the extended gamma and beta functions to the matrix case, study their properties, obtain different integral representations, and establish the connection of these generalizations with other known special functions of matrix argument.
This paper is divided into seven sections.Section 2 deals with some well known definitions and results on matrix algebra, zonal polynomials and special functions of matrix argument.In Section 3, the extended matrix variate gamma function has been defined and its properties have been studied.Definition and different integral representations of the extended matrix variate beta function are given in Section 4. Some integrals involving zonal polynomials and generalized extended matrix variate beta function are evaluated in Section 5.In Section 6, the distribution of the sum of dependent generalized inverted Wishart matrices has been derived in terms of generalized extended matrix variate beta function.We introduce the generalized extended matrix variate beta distribution in Section 7.

Some known definitions and results
In this section we give several known definitions and results.We first state the following notations and results that will be utilized in this and subsequent sections.Let A = (a ij ) be an m × m matrix of real or complex numbers.Then, A denotes the transpose of A; tr(A) = a 11 + • • • + a mm ; etr(A) = exp(tr(A)); det(A) = determinant of A; A = spectral norm of A; A = A > 0 means that A is symmetric positive definite, 0 < A < I m means that both A and I m − A are symmetric positive definite, and A 1/2 denotes the unique positive definite square root of A > 0.
Several generalizations of the Euler's gamma function are available in the scientific literature.The multivariate gamma function, which is fre-

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Ingeniería y Ciencia quently used in multivariate statistical analysis, is defined by where the integration is carried out over m × m symmetric positive definite matrices.By evaluating the above integral it is easy to see that The multivariate generalization of the beta function is given by The generalized hypergeometric function of one matrix argument as defined by Constantine [9] and James [15] is where C κ (X) is the zonal polynomial of m×m complex symmetric matrix X corresponding to the ordered partition κ k and κ k denotes summation over all partitions κ.The generalized hypergeometric coefficient (a) κ used above is defined by where (a) r = a(a + 1) • • • (a + r − 1), r = 1, 2, . . .with (a) 0 = 1.The parameters a i , i = 1, . . ., p, b j , j = 1, . . ., q are arbitrary complex numbers.
No denominator parameter b j is allowed to be zero or an integer or halfinteger ≤ (m − 1)/2.If any numerator parameter a i is a negative integer, a 1 = −r, then the function is a polynomial of degree mr.The series converges for all X if p ≤ q, it converges for ||X|| < 1 if p = q + 1, and, unless it terminates, it diverges for all X = 0 if p > q.
If X is an m × m symmetric matrix, and R is an m × m symmetric positive definite matrix, then the eigenvalues of RX are same as those of R 1/2 XR 1/2 , where R 1/2 is the unique symmetric positive definite square root of R. In this case C κ (RX) = C κ (R 1/2 XR 1/2 ) and p F q (a 1 , . . ., a p ; b 1 , . . ., b q ; RX) Two special cases of (10) are the confluent hypergeometric function and the Gauss hypergeometric function denoted by Φ and F , respectively.They are given by The integral representations of the confluent hypergeometric function Φ and the Gauss hypergeometric function F are given by and for X < I m , where Re(a) > (m − 1)/2 and Re(c − a) > (m − 1)/2.
For properties and further results on these functions the reader is referred to Herz [13], Constantine [9], James [15], and Gupta and Nagar [10].
Definition 2.1.The extended matrix variate gamma function, denoted by Γ m (a; Σ), is defined by where Re(Σ) > 0 and a is an arbitrary complex number.
From (21), one can easily see that for Re(Σ) > 0 and H ∈ O(m), Γ m (a; HΣH ) = Γ m (a; Σ) thereby Γ m (a; Σ) depends on the matrix Σ only through its eigenvalues if Σ is a real matrix.
From the definition, it is clear that if Σ = 0, then for Re(a) > (m−1)/2, the extended matrix variate gamma function reduces to the multivariate gamma function Γ m (a).

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Ingeniería y Ciencia (a; Σ), is defined by where Σ > 0 and a is an arbitrary complex number.
Further, for Σ = I m , substituting X = Y 1/2 ZY 1/2 with the Jacobian J(Z → X) = det(Y ) −(m+1)/2 in (25) and applying (21), the expression for Γ (α,β) m (a; I m ) is derived as In the following theorem we establish a relationship between generalized extended gamma function of matrix argument and multivariate gamma function through an integral involving the generalized extended gamma function of matrix argument and zonal polynomials.
Theorem 3.2.For a symmetric positive definite matrix T of order m, Note that the above corollary gives an interesting relationship between the generalized extended gamma function of matrix argument and multivariate gamma function.Substituting s = (m + 1)/2, in (32), we obtain (a; Σ) by its integral representation given in (25), the left hand side integral in (33) is re-written as Now, integrating first with respect to Σ and then with respect to Z by using Lemma 2.2, we obtain the desired result.
Proof.Let Z, Y and Σ be symmetric positive definite matrices of order m.Further, let λ 1 , . . ., λ m be the characteristic roots of the matrix
Theorem 3.5.Suppose that σ 1 and σ n are the smallest and largest eigenvalues of the matrix Σ > 0. Then Proof.Note that and therefore Now, applying the above inequality in (25), and using the integral representation of the generalized extended gamma function given in (25), we obtain the desired result.
By Hölder's inequality, it is possible to obtain an interesting inequality that follows.

Generalized extended matrix variate Beta function
In this section, a matrix variate generalization of ( 6) is defined and several of its properties are studied.
where a and b are arbitrary complex numbers and Σ > 0.
Using Kummer's relation (16), the above expression can also be written as From (38), it is apparent that B Replacing the confluent hypergeometric function by its integral representation in (38), changing the order of integration, and integrating Z by using (22), we obtain Proof.Replacing B (α,β) m (a, b; Σ) by its equivalent integral representation given in (39) and changing the order of integration, the integral in (41) is rewritten as where we have used the result ing.cienc., vol.12, no.24, pp.51-82, julio-diciembre.2016.
Finally, evaluating (42) by using the definition of multivariate beta function, we obtain the desired result.
By letting s = (m + 1)/2, in (41), we obtain an interesting relation between the multivariate beta function and the generalized extended beta function of matrix argument.
Theorem 4.3.For a and b arbitrary complex numbers and Re(Σ) > 0, Proof.Noting that and substituting for B

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In the following theorem, we present an important inequality that shows how the extended matrix variate beta function decreases exponentially compared to the multivariate beta function.
Theorem 4.5.Suppose that σ 1 and σ n are the smallest and largest eigenvalues of the matrix Σ.Then Proof.Similar to the proof of Theorem 3.5.
The next result is obtained by applying the Minkowski inequality for determinants.The famous Minkowski inequality states that if A and B are symmetric positive definite matrices of order m, then Theorem 4.6.For the generalized extended beta function of matrix argument, we have (a, b + 1/m; Σ) by their respective integral representation, one obtains Now, by noting that det(Z) 1/m + det(I m − Z) 1/m ≤ 1 we obtain the desired result.
In the following theorem and three corollaries we give closed form representations of the integral (44) for k = 1 and k = 2. (a, b; Σ) by its equivalent integral representation given in (40) and changing the order of integration, the integral in (49) is rewritten as Now, evaluating the integral containing Σ using Lemma 2.1, we obtain Further, substituting (51) in ( 50) and integrating with respect to X by using Lemma 2.4, we obtain where Re(α − s) > k 1 + (m − 1)/2 and Re(β − s) > k 1 + (m − 1)/2.Now, substituting appropriately, we obtain the result.

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Ingeniería y Ciencia Proof.We obtain the desired result by summing (54) and (55) and using the result Proof.We obtain the desired result by using

Application to multivariate statistics
The Wishart distribution, which is the distribution of the sample variance covariance matrix when sampling from a multivariate normal distribution, is an important distribution in multivariate statistival analysis.Recently, Bekker et al. [2,3] and Bekker, Roux and Arashi [1] have used Wishart distribution in deriving a number of matrix variate distributions.Further, Bodnar, Mazur and Okhrin [5] have considered exact and approximate distribution of the product of a Wishart matrix and a Gaussian vector.The inverted Wishart distribution is widely used as a conjugate prior in Bayesian statistics (Iranmanesh et al. [14]).Knowledge of densities of functions of inverted Wishart matrices is useful for the implementation of several statistical procedures and in this regard we show that the distribution of the sum of dependent generalized inverted Wishart matrices can be written in terms generalized extended beta function of matrix argument.If W ∼ IW m (ν, Ψ), ν > m − 1, Ψ > 0, then its p.d.f. is given by , W > 0.
Proof.The expected value of C κ (S) is derived as