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Distributions on symmetric cones II: Beta-Riesz distribution 3 1 0 José A. D´ıaz-Garc´ıa ∗ 2 Department of Statistics and Computation n 25350 Buenavista, Saltillo, Coahuila, Mexico a J E-mail: [email protected] 9 1 ] T Abstract S This article derives several properties of the Riesz distributions, such as their cor- . h responding Bartlett decompositions, the inverse Riesz distributions and the distri- t bution of the generalised variance for real normed division algebras. In addition, a m introduce a kindofgeneralisedbeta distribution termedbeta-Riesz distributionfor real normed division algebras. Two versions of this distributions are proposed and [ some properties are studied. 1 v 1 Introduction 5 2 5 It is imminent the important role played by Wishart and beta distributions type I and II 4 in the context of multivariate statistics. In particular, the relationship between these two . 1 distributions to obtain the beta distribution in terms of the distribution of two Wishart 0 matrices. Faraut and Korányi (1994), and subsequently Hassairi et al. (2005), propose a 3 beta-Rieszdistribution,whichcontainsasaspecialcasetothebetadistributionobtainedin 1 terms of the distribution Wishart, which shall be named beta-Wishart, all this subjects in : v the context of simple Euclidean Jordan algebras. Such beta-Riesz distribution is obtained i analogously to the beta-Wishart distribution, but starting with a Riesz distribution. Re- X cently, D´ıaz-Garc´ıa (2012) proposes two versions of the Riesz distribution for real normed r a division algebras. Based in these last two versions of the Riesz distributions, it is possible to obtain two versions of the beta-Riesz distributions, which by analogy with the beta-Wishart distributions are termed beta-Riesz type I. As in classical beta-Wishart distribution, in addition it is feasible to propose two version for the beta-Riesz distribution type II. Each of the two versions for each beta-Riesz distributions of type I and II, (both versions for each) contain as particular cases to beta-Wishart distribution type I and beta-Wishart distribution type II, respectively. Thisarticlestudiestwoversionsforbeta-RieszdistributionstypeIandIIforrealnormed division algebras. Section 2 reviews some definitions and notation on real normed division algebras. Andalso,introducesothermathematicaltoolsastwodefinitionsofthegeneralised gamma function on symmetric cones, two Jacobians with respect to Lebesgue measure and someintegralresultsforrealnormeddivisionalgebras. Section3proposesdiverseproperties ∗Correspondingauthor Key words. Wishart distribution; Beta-Riesz distribution; Riesz distribution, generalised beta function anddistribution,real,complexquaternionandoctonionrandommatrices. 2000Mathematical SubjectClassification. Primary60E05, 62E15;secondary15A52 1 of two versions of the Riesz distributions as their Bartlett decompositions, inverse Riesz distributions and the distribution of the generalized variance. Section 4 introduces two generalisedbeta functions and, in terms ofthese, two beta-Riesz distributions of type I and II are obtained for real normed division algebras. Also, the relationship between the Riesz distributions and the beta-Riesz distributions are studied. This section concludes studying the eigenvalues distributions of beta-Riesz distributions type I and II in their two versions for real normed division algebras. 2 Preliminary results A detailed discussion of real normed division algebras may be found in Baez (2002) and Ebbinghaus et al. (1990). For convenience, we shall introduce some notation, although in general we adhere to standard notation forms. For our purposes: Let F be a field. An algebra A over F is a pair (A;m), where A is a finite-dimensional vector space over F and multiplication m : A A A is an F-bilinear map; that is, for all λ F, x,y,z A, × → ∈ ∈ m(x,λy+z) = λm(x;y)+m(x;z) m(λx+y;z) = λm(x;z)+m(y;z). Two algebras (A;m) and (E;n) over F are said to be isomorphic if there is an invertible map φ:A E such that for all x,y A, → ∈ φ(m(x,y))=n(φ(x),φ(y)). By simplicity, we write m(x;y)=xy for all x,y A. Let A be an algebra over F. Then A is said t∈o be 1. alternative if x(xy)=(xx)y and x(yy)=(xy)y for all x,y A, ∈ 2. associative if x(yz)=(xy)z for all x,y,z A, ∈ 3. commutative if xy =yx for all x,y A, and ∈ 4. unital if there is a 1 A such that x1=x=1x for all x A. ∈ ∈ If A is unital, then the identity 1 is uniquely determined. AnalgebraAoverFissaidtobeadivision algebra ifAisnonzeroandxy =0 x=0 A A ⇒ or y =0 for all x,y A. A ∈ The term“divisionalgebra”,comesfromthe followingproposition,whichshowsthat,in such an algebra, left and right division can be unambiguously performed. Let A be an algebra over F. Then A is a division algebra if, and only if, A is nonzero and for all a,b A, with b = 0 , the equations bx = a and yb = a have unique solutions A ∈ 6 x,y A. I∈n the sequel we assume F= and consider classes of division algebras over or “real ℜ ℜ division algebras” for short. We introduce the algebras of real numbers , complex numbers C, quaternions H and ℜ octonions O. Then, if A is an alternative real division algebra, then A is isomorphic to , ℜ C, H or O. Let A be a real division algebra with identity 1. Then A is said to be normed if there is an inner product (, ) on A such that · · (xy,xy)=(x,x)(y,y) for all x,y A. ∈ 2 If A is a real normed division algebra, then A is isomorphic , C, H or O. ℜ There are exactly four normed division algebras: real numbers ( ), complex numbers ℜ (C), quaternions (H) and octonions (O), see Baez (2002). We take into accountthat should be taken into account, , C, H and O are the only normed division algebras; furthermore, ℜ they are the only alternative division algebras. Let A be a division algebra over the real numbers. Then A has dimension either 1, 2, 4 or 8. In other branches of mathematics, the parameters α=2/β and t=β/4 are used, see Edelman and Rao (2005) and Kabe (1984), respectively. Finally, observe that is a real commutative associative normed division algebras, ℜ C is a commutative associative normed division algebras, H is an associative normed division algebras, O is an alternative normed division algebras. Let β be the set of all m n matrices ofrank m n overA with m distinct positive Lm,n × ≤ singular values, where A denotes a real finite-dimensional normed division algebra. Let Am×n be the set of all m n matrices over A. The dimension of Am×n over is βmn. Let A Am×n, then A∗ =A¯T×denotes the usual conjugate transpose. ℜ ∈ Table 1 sets out the equivalence between the same concepts in the four normed division algebras. Table 1: Notation Generic Real Complex Quaternion Octonion notation Semi-orthogonal Semi-unitary Semi-symplectic Semi-exceptional Vβ type m,n Orthogonal Unitary Symplectic Exceptional Uβ(m) type Symmetric Hermitian Quaternion Octonion Sβ hermitian hermitian m We denote by Sβ the realvector space of all S Am×m such that S=S∗. In addition, letPβ betheconeomfpositive definitematrices S A∈m×m. Thus,Pβ consistofallmatrices S=Xm∗X, with X Lβ ; then Pβ is an open su∈bset of Sβ . m ∈ m,n m m LetDβ bethediagonalsubgroupof β consistingofallD Am×m,D=diag(d ,...,d ). m Lm,m ∈ 1 m Let Tβ(m) be the subgroup of all upper triangular matrices T Am×m such that t = 0 U ∈ ij for 1<i<j m. For any m≤atrix X An×m, dX denotes the matrix of differentials (dx ). Finally, we ij definethemeasure orv∈olumeelement(dX)whenX Am×n,Sβ ,Dβ or β ,seeDumitriu ∈ m m Vm,n (2002) and D´ıaz-Garc´ıaand Gutiérrez-Jáimez (2011). If X Am×n then (dX) (the Lebesgue measure in Am×n) denotes the exterior product ∈ of the βmn functionally independent variables m n β (dX)= dx where dx = dx(k). ij ij ij i=1j=1 k=1 ^ ^ ^ IfS Sβ (orS Tβ(m)witht >0,i=1,...,m)then(dS)(theLebesguemeasurein ∈ m ∈ U ii Sβ orinTβ(m))denotestheexteriorproductoftheexteriorproductofthem(m 1)β/2+m m U − functionally independent variables, m m β (dS)= ds ds(k). ii ij i=1 i<jk=1 ^ ^ ^ 3 Observe, that for the Lebesgue measure (dS) defined thus, it is required that S Pβ , that is, S must be a non singular Hermitian matrix (Hermitian definite positive matr∈ix).m If Λ Dβ then (dΛ) (the Legesguemeasurein Dβ ) denotes the exteriorproduct ofthe ∈ m m βm functionally independent variables n β (dΛ)= dλ(k). i i=1k=1 ^ ^ If H β then 1 ∈Vm,n m n (H∗dH )= h∗dh . 1 1 j i i=1j=i+1 ^ ^ where H = (H∗ H∗)∗ = (h ,...,h h ,...,h )∗ Uβ(n). It can be proved that this 1| 2 1 m| m+1 n ∈ differentialformdoesnotdependonthe choiceoftheH matrix. Whenn=1; β defines 2 Vm,1 the unit sphere in Am. This is, of course, an (m 1)β- dimensional surface in Am. When n=m and denoting H by H, (HdH∗) is termed−the Haar measure on Uβ(m). 1 The surface area or volume of the Stiefel manifold β is Vm,n 2mπmnβ/2 Vol( β )= (H dH∗)= , (1) Vm,n ZH1∈Vmβ,n 1 1 Γβm[nβ/2] where Γβ [a] denotes the multivariate Gamma function for the space Sβ . This can be m m obtained as a particular case of the generalised gamma function of weight κ for the space Sβ with κ = (k ,k ,...,k ), k k k 0, taking κ = (0,0,...,0) and which m 1 2 m 1 ≥ 2 ≥ ··· ≥ m ≥ for Re(a) (m 1)β/2 k is defined by, see Gross and Richards (1987), m ≥ − − Γβ [a,κ] = etr A Aa−(m−1)β/2−1q (A)(dA) (2) m ZA∈Pβm {− }| | κ m = πm(m−1)β/4 Γ[a+k (i 1)β/2] i − − i=1 Y = [a]βΓβ [a], (3) κ m where etr()=exp(tr()), denotes the determinant, and for A Sβ · · |·| ∈ m m−1 q (A)= A km A ki−ki+1 (4) κ m i | | | | i=1 Y with A = (a ), r,s = 1,2,...,p, p = 1,2,...,m is termed the highest weight vector, see p rs Gross and Richards (1987). Also, Γβ [a] = etr A Aa−(m−1)β/2−1(dA) m ZA∈Pβm {− }| | m = πm(m−1)β/4 Γ[a (i 1)β/2], − − i=1 Y and Re(a)>(m 1)β/2. In other bran−ches of mathematics the highest weight vector q (A) is also termed the κ generalised power of A and is denoted as ∆ (A), see Faraut and Korányi (1994) and κ Hassairi and Lajmi (2001). Additional properties of q (A), which are immediate consequences of the definition of κ q (A) and the following property 1, are: κ 4 1. if λ ,...,λ , are the eigenvalues of A, then 1 m m q (A)= λki. (5) κ i i=1 Y 2. q (A−1)=q−1(A)=q (A), (6) κ κ −κ 3. if κ=(p,...,p), then q (A)= Ap, (7) κ | | in particular if p=0, then q (A)=1. κ 4. if τ =(t ,t ,...,t ), t t t 0, then 1 2 m 1 2 m ≥ ≥···≥ ≥ q (A)=q (A)q (A), (8) κ+τ κ τ in particular if τ =(p,p,...,p), then q (A) q (A)= Apq (A). (9) κ+τ κ+p κ ≡ | | 5. Finally, for B Am×m in such a manner that C=B∗B Sβ , ∈ ∈ m q (BAB∗)=q (C)q (A) (10) κ κ κ and q (B−1AB∗−1)=(q (C))−1q (A). (11) κ κ κ Remark 2.1. Let (Sβ ) denote the algebra of all polynomial functions on Sβ , and P m m (Sβ ) the subspace of homogeneous polynomials of degree k and let κ(Sβ ) be an irre- Pk m P m ducible subspace of (Sβ) such that P m (Sβ )= κ(Sβ ). Pk m P m κ XM Notethatq isahomogeneouspolynomialofdegreek,moreoverq κ(Sβ ),seeGross and Richards κ κ ∈P m (1987). In (3), [a]β denotes the generalised Pochhammer symbol of weight κ, defined as κ m [a]β = (a (i 1)β/2) κ − − ki i=1 Y m πm(m−1)β/4 Γ[a+k (i 1)β/2] i − − = iY=1 Γβ [a] m Γβ [a,κ] = m , Γβ [a] m where Re(a)>(m 1)β/2 k and m − − (a) =a(a+1) (a+i 1), i ··· − is the standard Pochhammer symbol. 5 An alternative definition of the generalised gamma function of weight κ is proposed by Khatri (1966), which is defined as Γβ [a, κ] = etr A Aa−(m−1)β/2−1q (A−1)(dA) (12) m − ZA∈Pβm {− }| | κ m = πm(m−1)β/4 Γ[a k (m i)β/2] i − − − i=1 Y ( 1)kΓβ [a] = − m , (13) [ a+(m 1)β/2+1]β κ − − where Re(a)>(m 1)β/2+k . 1 − Similarly,fromHerz (1955, p. 480)the multivariate beta function for the spaceSβ , can m be defined as β[b,a] = Sa−(m−1)β/2−1 I Sb−(m−1)β/2−1(dS) Bm Z0<S<Im| | | m− | = Ra−(m−1)β/2−1 I +R−(a+b)(dR) m ZR∈Pβm| | | | Γβ [a]Γβ [b] = m m , (14) Γβ [a+b] m whereR=(I S)−1 I,Re(a)>(m 1)β/2andRe(b)>(m 1)β/2,seeD´ıaz-Garc´ıaand Gutiérrez-Jáimez − − − − (2009b). Now, we show three Jacobians in terms of the β parameter, which are proposed as ex- tensionsofreal,complex orquaternioncases,seeD´ıaz-Garc´ıaand Gutiérrez-Jáimez(2011). Lemma 2.1. Let X and Y Pβ matrices of functionally independent variables, and let Y =AXA∗+C, where A ∈β m and C Pβ are matrices of constants. Then ∈Lm,m ∈ m (dY)= A∗Aβ(m−1)/2+1(dX). (15) | | Lemma 2.2 (Cholesky’s decomposition). Let S Pβ and T Tβ(m) with t > 0, ∈ m ∈ U ii i=1,2,...,m. Then m 2m tβ(m−i)+1(dT) if S=T∗T; ii (dS)= 2miY=m1tβ(i−1)+1(dT) if S=TT∗. (16) ii i=1 Lemma 2.3. Let S Pβ. Then, igYnoring the sign, if Y=S−1 ∈ m (dY)= S−β(m−1)−2(dS). (17) | | Next is stated a general result, that is useful in a variety of situations, which enable us to transform the density function of a matrix X Pβ to the density function of its ∈ m eigenvalues, see D´ıaz-Garc´ıaand Gutiérrez-Jáimez (2009b). Lemma 2.4. Let X Pβ be a random matrix with a density function f (X). Then the joint density function∈of thme eigenvalues λ ,...,λ of X is X 1 m πm2β/2+̺ m (λ λ )β f(HLH∗)(dH) (18) Γβm[mβ/2]i<j i− j ZH∈Uβ(m) Y 6 where L=diag(λ ,...,λ ), λ > >λ >0, (dH) is the normalised Haar measure and 1 m 1 m ··· 0, β =1; m, β =2; ̺= − 2m, β =4;  −4m, β =8. − Finally, let’s recall the multidimensional convolution theorem in terms of the Laplace transform. For this purpose, let’s use the complexification Sβ,C =Sβ +iSβ of Sβ . That m m m m is, Sβ,C consist of all matrices X (FC)m×m of the form Z = X+iY, with X,Y Sβ . It commes to X= Re(Z) and Y = I∈m(Z) as the real and imaginary parts of Z, respec∈tivemly. The generalised right half-plane Φβ =Pβ +iSβ inSβ,C consistsofall Z Sβ,C suchthat Re(Z) Pβ, see (Gross and Richamrds, 19m87, p.m801). m ∈ m ∈ m Definition 2.1. If f(X) is a functionof X Pβ , the Laplacetransformoff(X) is defined ∈ m to be g(T)= etr XZ f(X)(dX). (19) ZX∈Pβm {− } where Z Φβ . ∈ m Lemma 2.5. If g (Z) and g (Z) are the respective Laplace transforms of the densities 1 2 f (X) and g (Y) then the product g (Z)g (Z) is the Laplace transform of the convolution X Y 1 2 f (X) g (Y), where X ∗ Y h (Ξ)=f (X) g (Y)= f (Υ)g (Ξ Υ)(dΥ), (20) Ξ X ∗ Y Z0<Υ<Ξ X Y − with Ξ=X+Y and Υ=X. 3 Riesz distributions ThissectionshowstwoversionsoftheRieszdistributions(D´ıaz-Garc´ıa,2012)andthestudy of their Bartlett decompositions. Also, inverse Riesz distributions are obtained. Definition 3.1. Let Σ Φβ and κ=(k ,k ,...,k ), k k k 0. ∈ m 1 2 m 1 ≥ 2 ≥···≥ m ≥ 1. Then it is said that X has a Riesz distribution of type I if its density function is βam+Pmi=1ki etr Σ−1X Xa−(m−1)β/2−1q (X)(dX) (21) Γβ [a,κ]Σaq (Σ) {− }| | κ m κ | | for Re(a) (m 1)β/2 k ; denoting this fact as X Rβ,I(a,κ,Σ). ≥ − − m ∼ m 2. Then it is said that X has a Riesz distribution of type II if its density function is qκ(Σ)βam−Pmi=1ki etr Σ−1X Xa−(m−1)β/2−1q (X−1)(dX) (22) Γβ [a, κ]Σa {− }| | κ m − | | for Re(a)>(m 1)β/2+k ; denoting this fact as X Rβ,II(a,κ,Σ). − 1 ∼ m Theorem 3.1. Let Σ Φβ and κ = (k ,k ,...,k ), k k k 0. And let Y =X−1. ∈ m 1 2 m 1 ≥ 2 ≥ ··· ≥ m ≥ 7 1. Then if X has a Riesz distribution of type I the density of Y is βam+Pmi=1ki etr Σ−1Y−1 Y−(a+(m−1)β/2+1)q (Y−1)(dY) (23) Γβ [a,κ]Σaq (Σ) {− }| | κ m κ | | for Re(a) (m 1)β/2 k and is termed as inverse Riesz distribution of type I. m ≥ − − 2. Then if X has a Riesz distribution of type II the density of Y is qκ(Σ)βam−Pmi=1ki etr Σ−1Y−1 Y−(a+(m−1)β/2+1)q (Y)(dY) (24) Γβ [a, κ]Σa {− }| | κ m − | | for Re(a) > (m 1)β/2+k and it said that Y has a inverse Riesz distribution of 1 − type II. Proof. It is immediately noted that(dX)= Y−β(m−1)−2(dY) andfrom (21) and(22). | | Observe that, if κ = (0,0,...,0) in two densities in Definition 3.1 and Theorem 3.1 the matrixmultivariategammaandinversegammadistributionsareobtained. Asconsequence, in this last case if Σ = 2Σ and a = βn/2, the Wishart and inverse Wishart distributions are obtained, too. Theorem 3.2. Let T Tβ(m) with t >0, i=1,2,...,m and define X=T∗T. ∈ U ii 1. If X has a Riesz distribution of type I, (21), with Σ=I , then the elements t (1 m ij i j m) of T are all independent. Furthermore, t2 β(a+k (i 1)β/2,1≤) ≤ ≤ ii ∼ G i − − and √2t β(0,1) (1 i<j m). ij ∼N1 ≤ ≤ 2. If X has a Riesz distribution of type II, (22), with Σ=I , then the elements t (1 m ij i j m) of T are all independent. Moreover, t2 β(a k (i 1)β/2,1) an≤d ≤ ≤ ii ∼ G − i− − √2t β(0,1) (1 i<j m). ij ∼N1 ≤ ≤ Wherex β(a,α)denotesagammadistributionwithparametersaandαandy β(0,1) ∼G ∼N1 denotes a random variable with standard normal distribution for real normed division algebras. Moreover, their respective densities are 1 β(x:a,α)= exp βx/α xa−1(dx), G (α/β)aΓ[a] {− } and 1 β(y :0,1)= exp βy2/2 (dy) N1 (2π/β)β/2 {− } where Re(a)>0 and α Φβ, see D´ıaz-Garc´ıa and Gutiérrez-Jaímez (2011). ∈ 1 Proof. ThisisgivenforthecaseofRieszdistributiontypeI.TheproofforRieszdistribution type II is the same thing. The density of X is βam+Pmi=1ki etr X Xa−(m−1)β/2−1q (X)(dX). (25) Γβ [a,κ] {− }| | κ m 8 Since X=T∗T we have m trX = trT∗T= t2 , ij i≤j X m X = T∗T = T2 = t2, | | | | | | ii i=1 Y m−1 m q (X) = q (T∗T)= T∗Tkm T∗T ki−ki+1 = t2ki, κ κ | | | i i| ii i=1 i=1 Y Y and by Theorem 2.2 noting that dt2 =2t dt , then ii ii ii m (dX) = 2m tβ(m−i)+1 dt , ii  ij i=1 i≤j Y ^   m = t2 β(m−i)/2 dt2 dt . ii ii!∧ ij i=1 i=1 i<j Y(cid:0) (cid:1) ^ ^   Substituting this expression in (25) and using (5) we find that the joint density of the t (1 i j m) can be written as ij ≤ ≤ ≤ m βa+ki−(i−1)β/2 exp βt2 t2 a+ki−(i−1)β/2−1 dt2 Γ[a+k (i 1)β/2] {− ii} ii ii i i=1 − − Y (cid:0) (cid:1) (cid:0) (cid:1) m 1 exp βt2 (dt ), × (π/β)β/2 {− ij} ij i<j Y only observe that βam+Pmi=1ki βam+Pmi=1ki−m(m−1)β/4 = Γβm[a,κ] β−m(m−1)β/4πm(m−1)β/4 mi=1Γ[a+ki−(i−1)β/2] m βa+ki−(i−1)β/2 Qm 1 = . Γ[a+k (i 1)β/2] (π/β)β/2 i i=1 − − i<j Y Y In analogy to generalisedvariance for Wishart case, the following result gives the distribution of X when X has a Riesz distribution type I or type II. | | Theorem 3.3. Let v = X/Σ. Then | | | | 1. if X has a Riesz distribution of type I, (21), the density of v is m β t2 :a+k (i 1)β/2,1 . G ii i− − i=1 Y (cid:0) (cid:1) 2. if X has a Riesz distribution of type II, (22), the density of v is m β t2 :a k (i 1)β/2,1 . G ii − i− − i=1 Y (cid:0) (cid:1) 9 where t2, i=1,...,m, are independent random variables. ii Proof. This is immediately from Theorem 3.2, noting that if B=Σ−1/2XΣ−1/2 =T∗T, with T Tβ(m) and t >0, i=1,2,...,m, then ∈ U ii m B = t2 = X/Σ =v. | | ii | | | | i=1 Y 4 Generalised beta distributions: Beta-Riesz distributions. Thissectiondefinesseveralversionsforthebetafunctionsandtheirrelationwiththegamma functionstypeIandII.Intheseterms,thebeta-RieszdistributionstypeIandIIaredefined. Finally, diverse properties are studied. 4.1 Generalised c-beta function A generalised of multivariate beta function for the cone Pβ , denoted as β[a,κ;b,τ], can m Bm be defined as Sa−(m−1)β/2−1q (S)I Sb−(m−1)β/2−1q (I S)(dS) (26) κ m τ m Z0<S<Im| | | − | − where κ = (k ,k ,...,k ), k k k 0, τ = (t ,t ,...,t ), t t 1 2 m 1 2 m 1 2 m 1 2 ≥ ≥ ··· ≥ ≥ ≥ ≥ ··· ≥ t 0, Re(a) > (m 1)β/2 k and Re(b) > (m 1)β/2 t . This is defined by m m m ≥ − − − − Faraut and Korányi (1994, p. 130) for Euclidean simple Jordan algebras. In the context of multivariateanalysis,thisgeneralisedbetafunctioncanbetermedgeneralisedc-betafunction type I, as analogy to the correspondence case of matrix multivariate beta distribution, and using the term c-beta as abbreviation of classical-beta. In the next theorem we introduce thegeneralised c-beta function type II anditsrelationwiththegeneralisedgammafunction. Theorem 4.1. The generalised c-beta function type I can be expressed as Ra−(m−1)β/2−1q (R)I +R−(a+b)q−1 (I +R)(dR) ZR∈Pβm| | κ | m | κ+τ m Γβ [a,κ]Γβ [b,τ] = m m , Γβ [a+b,κ+τ] m where κ = (k ,k ,...,k ), k k k 0, τ = (t ,t ,...,t ), t t 1 2 m 1 2 m 1 2 m 1 2 ≥ ≥ ··· ≥ ≥ ≥ ≥ ··· ≥ t 0, Re(a)>(m 1)β/2 k and Re(b)>(m 1)β/2 t . The integral expression is m m m ≥ − − − − termed generalised c-beta function type II. Proof. Let R = (I S)−1 I then by Lemma 2.3 (dS) = (I+R)−β(m−1)−2(d(R)). The desired result is ob−tained m−aking the change of variable on (26) noting that S = (I + m R)−1/2R(I +R)−1/2, then m S = R I +R−1, m | | | || | I S = I +R−1, m m | − | | | 10