chapter 5 fractional integral operators with many ...

CHAPTER 5 FRACTIONAL INTEGRAL OPERATORS WITH MANY SCALAR VARIABLES AND STATISTICAL DISTRIBUTIONS

5.0. Introduction We will consider the situtation of many real scalar variables first. It is always easier to introduce the subject matter through statistical distribution theory and Kober fractional integral operators. Hence this connection between statistical distributions and fractional integrals will be exploited first. This will enable us to come up with a general definition for fractional integrals of the first and second kind as Mellin convolutions of ratios and products in the real scalar variable case and as M-convolutions of ratios and products in the matrix-variate case. Here we will start with the investigation of products and ratios of real positive random variables. The cases involving two real scalar variables and two real matrix variables are already discussed in earlier chapters. Here we will consider many real scalar variables first. When going from a one-variable function to many-variable function there is no unique one to one correspondence. Many types of multivariable functions can be considered when one has the preselected one-variable function. Hence there is nothing called the multivariable analogue of a univariable operator. Hence we construct one multivariable operator here which is analogous to a one variable Kober fractional integral operator of the second and first kinds. Other such analogues can be defined. The discussions in the following sections will go parallel to the author’s papers in Cornell University arXiv. 5.1. Kober Fractional Integral Operators of the Second Kind 111

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First, we will establish the connection between Kober integral operators and statistical densities. Let x1 , x2 , ..., xk be independently distributed type-1 beta random variables with parameters (βj + 1, αj ), j = 1, ..., k, βj > −1, αj > 0, j = 1, ..., k. Usually the parameters in a statistical density are real but the following integrals also exist for complex parameters and in that case the conditions will be −1 and 0, j = 1, ..., k. That is, the density of xj is of the form fj (xj ) =

Γ(αj + βj + 1) βj x (1 − xj )αj −1 , 0 ≤ xj ≤ 1 Γ(αj )Γ(βj + 1) j

(5.1.1)

for αj > 0, βj > −1 and fj (xj ) = 0 elsewhere, j = 1, ..., k so that the joint density of x1 , ..., xk is the product f1 (x1 )...fk (xk ). Let y1 , ..., yk be another sequence of real scalar positive random variables having an arbitrary joint density f (y1 , ..., yk ). Let the two sets (x1 , ..., xk ) and (y1 , ..., yk ) be independently distributed. Consider the transformation uj = xj yj , vj = yj u or xj = yjj , yj = vj and the Jacobian of the transformation is

{

k Y j=1

∧dxj } ∧ {

k Y j=1

∧dyj } = {

k Y

−1

(vj )

}{

j=1

k Y j=1

∧duj } ∧ {

k Y

∧dvj }

(5.1.2)

j=1

Then the joint density of u1 , ..., uk , denoted by g(u1 , ..., uk ), is given by k k Z Y Γ(αj + βj + 1) Y 1 uj βj uj g(u1 , ..., uk ) = { }{ ( ) (1 − )αj −1 } Γ(βj + 1)Γ(αj ) j=1 vj vj vj vj j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk

Z k k Y Γ(αj + βj + 1) Y βj ∞ −β −α ={ }{ uj (vj − uj )αj −1 vj j j } Γ(β + 1)Γ(α ) j j vj =uj j=1 j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk .

(5.1.3)

Therefore we have the following result: Theorem 5.1.1. Let xj , yj , uj , vj , j = 1, ..., k be as defined above where x1 , ..., xk are independently type-1 beta distributed with parameters (βj +

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1, αj ), j = 1, ..., k, y1 , ..., yk having a joint arbitrary density f (y1 , ..., yk ) with (x1 , ..., xk ) and (y1 , ..., yk ) independently distributed. If the joint density of u1 , ..., uk is denoted as g(u1 , ..., uk ) then k Y

Γ(β + 1) −α ,j=1,...,k }g(u1 , ..., uk ) = K2,ujj,βj ,j=1,...,k f (v1 , ..., vk ) Γ(αj + βj + 1) j=1

{

(5.1.4)

−α ,j=1,...,k

for −1, 0, j = 1, ..., k., where K2,ujj,βj ,j=1,...,k f is the Kober fractional integral operator of the second kind of orders αj and parameters βj , j = 1, ..., k. Definition 5.1.1. Kober fractional integral operator of the second kind of orders αj , j = 1, ..., k and parameters βj , j = 1, ..., k in the multivariable case This will be defined as the following fractional integral and denoted as follows: −α ,j=1,...,k

K2,ujj,βj ,j=1,...,k f (v1 , ..., vk ) β k k Z ∞ Y Y uj j −β −α ={ }{ (vj − uj )αj −1 vj j j } Γ(αj ) j=1 vj =uj j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk .

(5.1.5)

This author has given the above definition, which is parallel to the one in the one variable case. We will now look at various connections to different problems. First we will establish a number of results in connection with statistical distribution theory. It is already shown above in (5.1.4) that (5.1.5) can be treated as a constant multiple of a joint density of a number of random variables u1 , ..., uk appearing in different contexts. In this case we had x1 , ..., xk mutually independently distributed and thus there were a total of k + 1 densities involved, the k of x1 , ..., xk and the one of (y1 , ..., yk ). Let us see what happens if x1 , ..., xk are not independently distributed but they have a joint density f1 (x1 , ..., xk ) and (y1 , ..., yk ) having a joint density f2 (y1 , ..., yk ). Then we can show that if f1 can be eventually reduced to independent type-1 beta form, still we can consider Kober operators of the second kind in the multivariable case as constant multiples of statistical densities.

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Let (x1 , ..., xk ) have a joint type-1 Dirichlet density with parameters (α1 + 1, ..., αk + 1; αk+1 ), −1, j = 1, ..., k, 0, that is, Γ(α1 + ... + αk+1 + k) α1 k f1 (x1 , ..., xk ) = Qk x1 ...xα k { j=1 Γ(αj + 1)}Γ(αk+1 ) × (1 − x1 − ... − xk )αk+1 −1 , 0 ≤ xj ≤ 1, j = 1, ..., k, 0 ≤ x1 + ... + xk ≤ 1

(5.1.6)

and f1 (x1 , ..., xk ) = 0 elsewhere. Let us consider the transformations x1 = t1 , x2 = t2 (1 − t1 ), ..., xk = (1 − t1 )...(1 − tk−1 ) or xj = tj (1 − t1 )(1 − t2 )...(1 − tj−1 ), j = 1, ..., k or tj =

xj , j = 1, ..., k. 1 − x1 − ... − xj−1

(5.1.7)

Under this transformation the Jacobian is (1 − t1 )k−1 ...(1 − tk−1 ). It is easy to show that under this transformation t1 , ..., tk will be independently distributed as type-1 beta variables with the parameters (αj + 1, βj ) with βj = αj+1 + αj+2 + ... + αk + (k − j) + αk+1 or tj has the density fj (tj ) =

Γ(αj + 1 + βj ) αj t (1 − tj )βj −1 , 0 ≤ tj ≤ 1 Γ(αj + 1)Γ(βj ) j

and fj (tj ) = 0 elsewhere, αj > −1, βj > 0, j = 1, ..., k. In the light of these observations, let us consider two sets of positive random variables (x1 , ..., xk ) and (y1 , ..., yk ) where the two sets are independently distributed with (x1 , ..., xk ) having a type-1 Dirichlet distribution. Let uj = tj vj = xj vj ( 1−x1 −...−x ), j = 1, ..., k. Then following through the same procedure j−1 as above we have the following theorem. Theorem 5.1.2. Let (x1 , ..., xk ) and (y1 , ..., yk ) be two sets of real scalar positive random variables where between sets they are independently distributed. Let (y1 , ..., yk ) have an arbitrary joint density f (y1 , ..., yk ) and let (x1 , ..., xk ) have a type-1 Dirichlet density with the parameters (α1 + 1, ..., αk + 1; αk+1 ) or with the density αk αk+1 −1 1 f1 (x1 , ..., xk ) = C xα 1 ...xk (1 − x1 − ... − xk ) 0 ≤ xj ≤ 1, 0 ≤ x1 + ... + xk ≤ 1, j = 1, ..., k

(5.1.8)

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and f1 (x1 , ..., xk ) = 0 elsewhere, where C is the normalizing constant. Let xj uj = vj ( 1−x1 −...−x ), yj = vj , j = 1, ..., k. If the joint density of u1 , ..., uk j−1 is again denoted by g(u1 , ..., uk ) then k Y

Γ(αj + 1) −α ,j=1,...,k }g(u1 , .., uk ) = K2,ujj,βj ,j=1,...,k f (v1 , ..., vk ) Γ(α + β + 1) j j j=1

{

(5.1.9)

where βj = αj+1 + αj+2 + ... + αk + (k − j) + αk+1 , j = 1, ..., k, −1, j = 1, ..., k, 0, 0, j = 1, ..., k. The above structure indicates that we can consider any multivariable density f1 (x1 , ..., xk ) for a set of real scalar positive random variables (x1 , ..., xk ) and if we can find a suitable transformation to bring the joint density of the new variables as products of type-1 beta densities then the Kober fractional integral operator of the second kind for the multivariable case can be written in terms of a statistical density as shown above. There are many densities where a transformation can bring f1 (x1 , .., xk ) to product of type-1 beta densities. There are several generalizations of type-1 and type-2 Dirichlet models where suitable transformations exist which can bring a set of mutually independently distributed type-1 beta random variables. We will list one more example of this type before quitting this section. Let us consider a generalized type-1 Dirichlet model of the following type. Several types of generalizations of the following category are available. β1 α2 β2 1 f1 (x1 , ..., xk ) = C1 xα 1 (1 − x1 ) x2 (1 − x1 − x2 ) ... βk −1 k × xα , k (1 − x1 − ... − xk ) 0 ≤ xj ≤ 1, 0 ≤ x1 + ... + xj ≤ 1, j = 1, ..., k (5.1.10)

and f1 (x1 , ..., xk ) = 0 elsewhere, where C1 is a normalizing constant. Let us consider the same transformation as in (5.1.7). Then we can show that t1 , ..., tk will be mutually independently distributed as type-1 beta random variables with the parameters (αj + 1, γj ), j = 1, ..., k where γj = αj+1 + αj+2 + ... + αk−1 + βj + βj+1 + ... + βk + (k − j)

(5.1.11)

for j = 1, ..., k with −1, j = 1, ..., k and 0, j = 1, ..., k.

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Theorem 5.1.3. Let x1 , ..., xk have a joint density of the form in (5.1.10). Let y1 , .., yk be another set of real scalar positive random variables having an arbitrary density f (y1 , ..., yk ). Between sets let (x1 , ..., xk ) and (y1 , ..., yk ) be independently distributed. Consider the transformation as in (5.1.7) where let xj ), yj = vj , j = 1, ..., k. uj = vj ( 1 − x1 − ... − xj−1 Let the joint density of u1 , ..., uk be again denoted by g(u1 , ..., uk ). Then Y {

Γ(αj + 1) }g(u1 , ..., uk ) Γ(αj + γj + 1) −α ,j=1,...,k

= K2,ujj,γj ,j=1,...k f (v1 , ..., vk )

(5.1.12)

where γj = αj+1 + αj+2 + ... + αk + βj + βj+1 + ... + βk + (k − j), j = 1, ..., k for −1, 0, j = 1, ..., k. 5.1.2. A Pathway Generalization of Kober Operator of the Second Kind in the Multivariable Case Let x1 , ..., xk be independently distributed with xj having a pathway density given by fj (xj ) =

β cjp xj j [1

− aj (1 − qj )xj ]

ηj 1−qj

(5.1.13)

for 1 − aj (1 − qj )xj > 0, aj > 0, qj < 1, ηj > 0, βj > −1 and fj (xj ) = 0 elsewhere, where cjp =

[aj (1 − qj )]βj +1 Γ(βj + 1 + η

ηj 1−qj

Γ(βj + 1)Γ( 1−qj j + 1)

+ 1)

.

(5.1.14)

Let y1 , ..., yk be real scalar positive random variables with a joint density f (y1 , ..., yk ). Let (x1 , .., xk ) and (y1 , ..., yk ) be statistically independently u distributed. Let uj = xj yj , vj = yj , xj = vjj , j = 1, ..., k. Then the Jacobian of the transformation is (v1 ...vk )−1 . Let the joint density of u1 , ..., uk be denoted again by g(u1 , ..., uk ). Then from the standard technique of transformation of variables the density g is given by

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g(u1 , ..., uk ) = {

k Y

Z β cjp uj j

η

∞

j −βj −( 1−q +1)

vj =aj (1−qj )uj

j=1

117

vj

j

ηj

× [vj − a(1 − qj )uj ] 1−qj }f (v1 , ..., vk ) × dv1 ∧ ... ∧ dvk .

(5.1.15)

Hence we may define a pathway extension of Kober fractional integral operator of the second kind in the multivariable case. Definition 5.1.2. A Pathway Kober Fractional Integral Operator of the Second Kind for the Multivariable Case It will be defined and denoted as follows: η

j −( 1−q +1),j=1,...,k

K2,uj ,βjj,aj ,qj ,j=1,...,k f (v1 , ..., vk ) Z β k ηj Y uj j 1−qj } (v − aj (1 − qj )uj ) ={ η Γ( 1−qj j + 1) vj >aj (1−qj )uj j=1 × f (v1 , ..., vk )dv1 ∧ ... ∧ dvk .

(5.1.16)

for aj > 0, qj < 1, ηj > 0, −1. Theorem 5.1.4. Let x1 , ..., xk , y1 , ..., yk , uj , j = 1, ..., k and g(u1 , ..., uk ) be as defined in (5.1.15). Let the pathway extended Kober operator be as defined in (5.1.16). Then k Y

Γ(βj + 1) }g(u1 , ..., uk ) η Γ(βj + 1−qj j + 2) j=1

{

η

j +1),j=1,...,k −( 1−q

= K2,uj ,βjj,aj ,qj ,j=1,...,k f (v1 , ..., vk ). (5.1.17) When any particular qr → 1− then we can see the corresponding factor goes to the exponential form. ηr (1 − qr )βr +1 Γ(βr + 1−q + 1) ur β 1 ur ηr r lim ( ) r [1 − ar (1 − qr )( )] 1−qr ηr qr →1− Γ( 1−qr + 1) vr vr vr =(

ur βr 1 −ar ηr ) e vr vr

r (u vr )

, 0 ≤ vr < ∞.

(5.1.18)

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Thus, individual qj ’s can go to 1 and the corresponding factor will go to exponential form or the correspondingly we get a gamma density structure for that factor. Exercises 5.1. 5.1.1. Evaluate the Kober fractional integral operator of the second kind of (5.1.5) for the power function or when f (v1 , ..., vk ) = v1±δ1 ...vk±δk and show that it is k Y j=1

±δj

uj

Γ(βj ± δj ) , 0, 0, j = 1, ..., k. Γ(αj + βj ± δj )

5.1.2. Taking   k Y αj −1  (1 − xj ) f1 (x1 , ..., xk ) = φ1 (x1 , ..., xk ) , 0, j = 1, ..., k   Γ(αj ) j=1

and f2 (y1 , ..., yk ) = φ2 (y1 , ..., yk )f (y1 , ..., yk ) where φ1 and φ2 are specified functions and f is an arbitrary function, and taking the product uj = xj yj , j = 1, ..., k define Mellin convolution of this product, denoted by g(u1 , ..., uk ), as    Z Y k  1 u1 1 uj αj −1  uk φ1 ( , ..., ) g(u1 , ..., uk ) = (1 − )   Γ(αj )  v j=1 vj vj v1 vk j=1  k Y

× φ2 (v1 , ..., vk )f (v1 , ..., vk )dv1 ∧ ... ∧ dvk , 0, j = 1, ..., k. Define g(u1 , ..., uk ) as the fractional integral of the second kind of orders −α ,j=1,...,k αj , j = 1, ..., k D2,ujj,j=1,...,k f . Specify φ1 and φ2 , where f is arbitrary, so that one has Kober fractional integral of the second kind of order αj , j = 1, ..., k defined in Seciton 5.1.

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5.1.3. Define Riemann-Liouville fractional integral of the second kind of orders αj , j = 1, ..., k in the multivariable case or −α ,j=1,...,k

−α ,j=1,...,k

D2,ujj,j=1,...,k f = D2,(ujj ,bj ),j=1,...,k f   k Z bj  Y αj −1 f (v1 , ..., vk )dv1 ∧ ... ∧ dvk , (vj − uj ) =   vj =uj j=1

for 0, j = 1, ..., k. Specify φ1 and φ2 in Exercise 5.1.2 and derive this Riemann-Liouville fractional integral in the multivariable case as a Mellin convolution of a product. 5.1.4. Define Weyl fractional integral of the second kind of orders αj , j = 1, ..., k as   k Z ∞ Y αj −1  (vj − uj ) −α ,j=1,...,k W2,ujj,j=1,...,k f = f (v1 , ..., vk )dv1 ∧ ... ∧ dvk   Γ(αj ) j=1 uj for 0, j = 1, ..., k. Specify φ1 and φ2 in Exercise 5.1.2 and derive this Weyl fractional integral of the second kind of orders αj , j = 1, ..., k. 5.1.5. Evaluate the Riemann-Liouville operator in Exercise 5.1.3 and the Weyl operator in Exercise 5.1.4 when f (v1 , ..., vk ) =

k Y

e−aj vj , aj > 0, vj > 0, j = 1, ..., k.

j=1

5.2. Mellin Transform in the Multivariable Case for Kober Operators of the Second Kind The Mellin transform in the multivariable case is defined as M {f (x1 , ..., xk ); s1 , ..., sk }

Z =

Z

∞

... 0

0

∞

xs11 −1 ...xskk −1

× f (x1 , ..., xk )dx1 ∧ ... ∧ dxk

(5.2.1)

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whenever it exists, where s1 , ..., sk in general are complex parameters. Hence for the Kober operator of the second kind we have −α ,j=1,...,k

M {K2,ujj,βj ,j=1,...,k f (v1 , ..., vk ); s1 , ..., sk } Z ∞ Z ∞ Z β k Y uj j −β−αj sk −1 s1 −1 = ... u1 ...uk { (vj − uj )αj −1 vj } Γ(αj ) vj >uj 0 0 j=1 × f (v1 , ..., vk )dV dU where, for example, dU = du1 ∧ ... ∧ duk Z ∞ Z ∞ k Y −β −α = ... f (v1 , ..., vk ){ vj j j } 0

× [{

k Y

j=1

0

Z 0

vj

j=1 s +βj −1

uj j

Take out vj , put yj =

uj vj

(vj − uj )αj −1 duj }]dV.

then the integral over uj will go to

β +αj +sj −1 Γ(αj )Γ(βj

vj j

+ sj ) . Γ(αj + βj + sj )

Hence the required Mellin transform is   k Y Γ(βj + sj )  ∗  f (s1 , ..., sk ) Γ(α + β + s ) j j j j=1 where f ∗ is the Mellin transform of f . Then we have the following theorem. Theorem 5.2.1. For the Kober fractional integral operator of the second kind in the multivariable case as defined in (5.1.5) the Mellin transform, with Mellin parameters s1 , ..., sk , is given by −α ,j=1,...,k

M {K2,ujj,βj ,j=1,...,k f (v1 , ..., vk ); s1 , ..., sk }   k Y Γ(βj + sj )  f ∗ (s1 , ..., sk ) (5.2.2) = Γ(α + β + s ) j j j j=1

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for 0, 0, j = 1, ..., k. Exercises 5.2. 5.2.1. Evaluate the multivariable Mellin transform in each case of Exercises 5.1.1-5.1.5, denoting the Mellin transform of f (v1 , ..., vk ) as f ∗ (s1 , ..., sk ) where s1 , ..., sk are Mellin parameters. Evaluate explicitly for the cases k = 1, k = 2, k = 3. 5.3. Kober Fractional Integral Operator of the First Kind for Multivariable Case In the multivariable case we will start with the following definition and notation. Definition 5.3.1. Kober Fractional Integral Operator of the First Kind in the Multivariable Case −α ,j=1,...,k K1,ujj,βj ,j=1,...,k f (v1 , ..., vk )

−β −α Z k Y u j j j uj β ={ (uj − vj )αj −1 vj j } Γ(α ) j vj =0 j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk .

(5.3.1)

First, we will derive this operator as a constant multiple of a statistical density. To this end, let x1 , ..., xk be independently distributed type-1 beta random variables with the parameters (βj , αj ), j = 1, ..., k or with the density fj (xj ) =

Γ(βj + αj ) βj −1 x (1 − xj )αj −1 , 0 ≤ xj ≤ 1, Γ(βj )Γ(αj ) j

(5.3.2)

for αj > 0, βj > 0 or 0, 0 when the parameters are in the complex domain. Let (y1 , ..., yk ) be real scalar positive random variables y having a joint density f (y1 , ..., yk ). Let yj = vj , uj = xjj , j = 1, ..., k. The Jacobian is given by dX ∧ dY = [

k Y j=1

(−

vj )]dU ∧ dV u2j

(5.3.3)

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where dX = dx1 ∧ ... ∧ dxk , dY = dy1 ∧ ... ∧ dyk . Similar wedge products for dU and dV . The joint density of u1 , ..., uk , following through the earlier steps, denoted by g1 (u1 , ..., uk ), is given by Z k k Y Γ(βj + αj ) Y −βj −αj uj β g1 (u1 , ..., uk ) = { uj (uj − vj )αj −1 vj j } }{ Γ(β )Γ(α ) j j v=0 j=1 j=1 × f (v1 , ..., vk )dv1 ∧ ... ∧ dvk .

(5.3.4)

Hence we have the following theorem. Theorem 5.3.1. Let x1 , ..., xk be independently distributed type-1 beta random variables and let y1 , ..., yk , v1 , ..., vk , u1 , ..., uk be as defined in (5.3.2) and (5.3.3). Let the joint density of u1 , ..., uk be denoted by g1 (u1 , ..., uk ). Then k Y

Γ(βj ) −α ,j=1,...,k }g1 (u1 , ..., uk ) = K1,ujj,βj ,j=1,...,k f (v1 , ..., vk ). Γ(βj + αj ) j=1

{

(5.3.5)

We can have pathway extension to Kober operator of the first kind, parallel to the results for the case of second kind. Other properties follow parallel to those for the case of the operator of the second kind. We will evaluate the multivariable Mellin transform following through steps parallel to those in the case of the second kind and hence we give the result here as a theorem. Theorem 5.3.2. The Mellin transform, with Mellin parameters s1 , ..., sk , for Kober fractional integral operator of the first kind in the multivariable case is given by the following: −α ,j=1,...,k

M {K1,ujj,βj ,j=1,...,k f (v1 , ..., vk ); s1 , ..., sk } k Y

Γ(1 + βj − s) }f ∗ (s1 , ..., sk ) Γ(1 + αj + βj − s) j=1

={

(5.3.6)

for 0, 0, 0, 0, j = 1, ..., k. From here we can have a definition for a pathway extension of Kober operator of the first kind in the multivariable case. Definition 5.3.2. Kober fractional integral operator of the first kind in the multivariable case Let the variables and parameters be as defined in Theorem 5.3.3. Then the pathway extended Kober operator of the first kind is defined and denoted as follows: η

j −( 1−q +1),j=1,...,k

K1,uj ,βjj,aj ,qj ,j=1,...,k f (v1 , ..., vk ) ηj uj +1) Z −βj −( 1−q k ηj Y j 1−aj (1−qj ) u β 1−qj ={ [uj − aj (1 − qj )vj ] vj j } ηj Γ( 1−qj + 1) vj =0 j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk for aj > 0, qj < 1, ηj > 0, 0, j = 1, ..., k. We can list several theorems when x1 , ..., xk are not independently distributed. Two such cases will be listed here without proofs. The proofs will be parallel to those in Section 5.1 and hence omitted.

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Theorem 5.3.4. Let x1 , ..., xk have a type-1 Dirichlet density as in (5.1.6) with αj replaced by αj −1 for j = 1, ..., k and let (y1 , ..., yk ) have an arbitrary joint density f (y1 , ..., yk ) where (x1 , ..., xk ) and (y1 , ..., yk ) are independently v xj and let uj = tjj , j = 1, ..., k. Let the joint distributed. Let tj = 1−x1 −...−x j−1 density of u1 , ..., uk be again denoted by g1 (u1 , ..., uk ). Let γj = αj+1 + αj+2 + ... + αk . Then −α ,j=1,...,k K1,ujj,γj ,j=1,...,k f (v1 , ..., vk )

={

k Y

Γ(γj ) }g1 (u1 , ..., uk ). Γ(αj + γj ) j=1

(5.3.9)

The next theorem is parallel to Theorem 5.1.3 for the operator of the second kind. Theorem 5.3.5. Let x1 , ..., xk have a joint density as in (5.1.10) with xj αj replaced by αj − 1, j = 1, ..., k. Let tj = 1−x1 −...−x as defined in j−1 (5.1.7). Let (y1 , ..., yk ) have an arbitrary joint density f (y1 , ..., yk ). Let v (x1 , ..., xk ) and (y1 , ..., yk ) be independently distributed. Let uj = tjj or v tj = ujj , j = 1, ..., k. Let γj = αj+1 + αj+2 + ... + αk + βj + βj+1 + ... + βk . Then −α ,j=1,...,k K1,ujj,γj ,j=1,...,k f (u1 , ..., uk )

k Y

Γ(αj ) }g1 (u1 , ..., uk ) Γ(α + γ ) j j j=1

={

for 0, 0, j = 1, ..., k. Exercises 5.3. 5.3.1. Let   k Y αj −1  (1 − xj ) f1 (x1 , ..., xk ) = φ1 (x1 , ..., xk )   Γ(αj ) j=1

and f (y1 , ..., yk ) = φ2 (y1 , ..., yk )f (y1 , ..., yk )

(5.3.10)

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where f (y1 , ..., yk ) is an arbitrary function and φ1 , φ2 are specified functions. Qk y v Let uj = xjj , yj = vj so that the Jacobian is j=1 (− uj2 ). Let the Mellin j

convolution of the ratio uj , j = 1, ..., k be denoted by g1 (u1 , ..., uk ). Then show that

g1 (u1 , ..., uk ) =

 k Z Y 

j=1

vj



vj αj −1  uj )

vj (1 − u2j Γ(αj )



vk v1 , ..., ) u1 uk × φ2 (v1 , ..., vk )f (v1 , ..., vk )dv1 ∧ ... ∧ dvk , 0, j = 1, ..., k. × φ1 (

5.3.2. Define Riemann-Liouville fractional integral of the first kind of orders αj , j = 1, ..., k in the multivariable case as −α ,j=1,...,k

−α ,j=1,...,k

D1,ujj,j=1,...,k f = D1,(ajj ,uj ),j=1,...,k f    k k    Y Y 1 = (uj − vj )α1 −1 f (v1 ...., vk )   Γ(αj )   j=1

j=1

× dv1 ∧ ... ∧ dvk . Specify φ1 and φ2 in Exercise 5.3.1 so that g1 (u1 , ..., uk ) agrees with RiemannLiouville fractional integral of the first kind defined here. 5.3.3. Define Weyl fractional integral of the first kind of orders αj , j = −α ,j=1,...,k 1, ..., k in the multivariable case, denoted by W1,ujj,j=1,...,k f as −α ,j=1,...,k E1,ujj,j=1,...,k f

=

−αj ,j=1,...,k W1,(0,u f j ),j=1,...,k

k Y (uj − vj )αj −1 ={ } Γ(αj ) j=1

× f (v1 , ..., vk )dv1 ∧ ... ∧ dvk . Specify φ1 , φ2 in Exercise 5.3.1 so that g1 (u1 , ..., uk ) agrees with the Weyl fractional integral of the first kind of order αj , j = 1, ..., k in the multivariable case defined here.

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5.3.4. Specify φ1 , φ2 in Exercise 5.3.1 so that g1 there agrees with the Kober fractional integral of the first kind and of orders αj , j = 1, ..., k defined in equation (5.3.1). 5.3.5. Evaluate the Mellin transforms of the fractional integral operators of the first kind, with Mellin parameters s1 , ..., sk , in Exercises 5.3.2, 5.3.3 and 5.3.4. Denote the Mellin transform of the arbitrary function f (v1 , ..., vk ) as f ∗ (s1 , ..., sk ). Evaluate explicitly for k = 1, k = 2, k = 3. 5.4. Fractional Integrals Involving Many Matrix Variables Let X1 , ..., Xk and Y1 , ..., Yk be two sets of p × p matrix random variables where between the sets the two sets are statistically independently distributed. Further, let Y1 , ..., Yk have an arbitrary joint density f (Y1 , ..., Yk ) and let X1 , ..., Xk be mutually independently distributed type-1 real matrixvariate beta random variables with the parameters (βj + p+1 2 , αj ), j = 1, ..., k. That is, Xj has the density

fj (Xj ) =

p+1 βj 2 ) p+1 |Xj | |I Γp (αj )Γp (βj + 2 )

Γp (αj + βj +

for O < Xj < I,

p−1 2 , −1 and fj (Xj ) = 0, j = 1, ..., k 1

1

elsewhere. Consider the transformation Uj = Yj2 Xj Yj2 , Yj = Vj , j = 1, ..., k −1

−1

or Xj = Vj 2 Uj Vj 2 , Yj = Vj , j = 1, ..., k, Here Uj can be called a symmetric product of Xj and Yj . We will be looking at the joint density of such product Uj , j = 1, ..., k. The Jacobian in the above transformation is p+1 p+1 |V1 |− 2 ...|Vk |− 2 . Substituting in (5.4.1) the joint density of the products U1 , ..., Uk , denoted by g2 (U1 , ..., Uk ), is given by k Y Γp (αj + βj +

p+1 2 ) { Γ (αj )Γp (βj + p+1 2 ) j=1 p

g2 (U1 , ..., Uk ) = Z |Vj |−

× Vj >Uj

p+1 2

− 12

|Vj

− 12 βj

Uj Vj

1

− 12 αj − p+1 2

| |I − V − 2 Uj Vj

× f (V1 , ..., Vk )dV1 ∧ ... ∧ dVk

|

}

Lecture Notes on Fractional Calculus k Y Γp (αj + βj +

p+1 βj 2 ) p+1 |Uj | Γ (αj )Γp (βj + 2 ) j=1 p

={ Z

127

|Vj |−βj −αj |Vj − Uj |αj −

×

p+1 2

}

Vj >Uj

× f (V1 , ..., Vk )dV1 ∧ ... ∧ dVk .

(5.4.2)

Hence we will define Kober fractional integral operator of the second kind and of orders (α1 , ..., αk ) and with parameters β1 , ..., βk for the many matrixvariate case, and denoted as follows: −α ,j=1,...,k

K2,Ujj ,βj ,j=1,...,k f (V1 , ..., Vk ) Z k Y p+1 |Uj |βj |Vj |−βj −αj |Vj − Uj |αj − 2 } ={ Γ (αj ) Vj >Uj j=1 p × f (V1 , ..., Vk )dV.

(5.4.3)

Therefore this Kober fractional integral operator of the second kind is a constant times a statistical density function of products of real matrixvariate random variables, namely, {

p+1 2 ) }g2 (U1 , ..., Uk ) βj + p+1 2 )

Γp (βj + Γp (αj +

−α ,j=1,...,k

= K2,Ujj ,βj ,j=1,...,k f (V1 , ..., Vk ). (5.4.4)

Now, let us consider f1 having a joint density of the matrix variables X1 , ..., Xk and f2 is a joint density of the matrix variables Y1 , ..., Yk where the two sets are independently distributed. Then we can have several interesting results where the Kober operator of (5.4.4) will become constant multiples of statistical densities coming from various considerations. Theorem 5.4.1. Let the two sets X1 , ..., Xk and Y1 , ..., Yk of matrices be statistically independently distributed. Let Y1 , ..., Yk have a joint arbitrary density f (Y1 , ..., Yk ). Further, let X1 , ..., Xk have a joint type-1 Dirichlet density with the parameters βj + p+1 2 , j = 1, ..., k, βk+1 . Consider the transformation X1 = T1

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A.M. Mathai 1

1

X2 = (I − T1 ) 2 T2 (I − T1 ) 2 1

1

1

1

Xj = (I − Tj−1 ) 2 ...(I − T1 ) 2 Tj (I − T1 ) 2 ...(I − Tj−1 ) 2 , j = 2, ..., k.

(5.4.5)

Or 1

1

Tj = (I − X1 − ... − Xj−1 )− 2 Xj (I − X1 − ... − Xj−1 )− 2 , j = 2, ..., k, T1 = X1 . (5.4.6) Consider the transformation 1

1

− 12

Yj = Vj , Uj = Vj2 Tj Vj2 , Tj = Vj

− 12

Uj Vj

, j = 1, ..., k.

(5.4.7)

Then the joint density of U1 , ..., Uk is constant times the generalized Kober operator of the second kind defined in (5.4.4). Proof: Under the transformation in (5.4.5) or (5.4.6) the Jacobian is J = |I − T1 |(k−1)(

p+1 2 )

|I − T2 |(k−2)(

p+1 2 )

...|I − Tk |

p+1 2

(5.4.8)

and that T1 , ..., Tk are independently distributed as type-1 real matrixvariate beta random variables with the parameters (βj + p+1 2 , γj ), j = 1, ..., k where γj = βj+1 + βj+2 + ... + βk + (k − j) (5.4.9) see, for example, Mathai (1997). Now, it is equivalent to the situation in (5.4.3) and (5.4.4) with Tj ’s standing in place of the independently distributed Xj ’s and hence from (5.4.4) we have the following result:

{

k Y j=1

p+1 2 ) }g2 (U1 , ..., Uk ) βj + p+1 2 )

Γp (βj + Γp (γj +

−γ ,j=1,...,k

= K2,Ujj ,βj ,j=1,...,k f (T1 , ..., Tk ) (5.4.10)

where γj is given in (5.4.9). Hence the result. We can consider several generalized models belonging to the family of generalized type-1 Dirichlet family in the many matrix-variate cases. In all such situations we can derive the generalized Kober fractional integral operator of the second kind in many matrices. We will take one such generalization here and obtain a theorem. Let f1 (X1 , ..., Xk ) be of the form

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129

f1 (X1 , ..., Xk ) = C |X1 |ζ1 |I − X1 |β1 |X2 |ζ2 |I − X1 − X2 |β2 ... × |Xk |ζk |I − X1 − ... − Xk |βk −

p+1 2

(5.4.11)

where O < X1 + ... + Xj < I, j = 1, ..., k, −1, j = 1, ..., k, C is the normalizing constant. Other conditions on the parameters will be given later. In this connection we can establish the following theorem. Theorem 5.4.2. Let X1 , ..., Xk have a joint density as in (5.4.11). Consider the transformation as in (5.4.5) and (5.4.6) with the Uj ’s and Vj ’s defined as in (5.4.7). Let the joint density of U1 , ..., Uk be denoted again as g2 (U1 , ..., Uk ). Let

δj = ζj+1 + ... + ζk + βj + βj+1 + ... + βk , j = 1, ..., k.

(5.4.12)

Then {

k Y j=1

p+1 2 ) }g2 (U1 , ..., Uk ) p+1 2 + δj )

Γp (ζj + Γp (ζj +

−δ ,j=1,..,k

= K2,Ujj ,ζj ,j=1,...,k f (V1 , ..., Vk ) (5.4.13)

where δj is defined in (5.4.12) and the Kober operator in (5.4.4). Proof: We can show that under the transformation in (5.4.5) or (5.4.6) the Tj ’s are independently distributed as real matrix-variate type-1 beta random variables with the parameters (ζj + p+1 2 , δj ), j = 1, ..., k. Now the result follows from the procedure of the proof in Theorem 5.4.1. For Kober operators in the one variable case see Mathai and Haubold (2008). Note 5.4.1. Special cases connecting to Riemann-Liouville operator, Weyl operator and Saigo operator, corresponding to the ones in Section 5.4.1 for Kober fractional integral operator of the second kind, can be obtained in a parallel manner and hence they will not be repeated here. Exercises 5.4.

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5.4.1. Let p+1 k Y |I − Xj |αj − 2 f1 (X1 , ..., Xk ) = φ1 (X1 , ..., Xk ){ } Γ (α ) p j j=1

and f2 (Y1 , ..., Yk ) = φ2 (Y1 , ..., Yk )f (Y1 , ..., Yk ) for p−1 2 , j = 1, ..., k, where φ1 and φ2 are specified functions and f is an arbitrary function. Consider the transformation or the matrix products − 12

1

1

Vj = Yj , Uj = Yj2 Xj Yj2 , Xj = Vj Show that (1): dXj = |Vj |− Jacobian in this case is

p+1 2

, j = 1, ..., k.

dUj , for fixed Vj , j = 1, ..., k; (2): the

k Y

dX1 ∧ ... ∧ dYk = {

− 12

Uj Vj

|Vj |−

p+1 2

}dU1 ∧ ... ∧ dVk .

j=1

Show that the M-convolution of the products U1 , j = 1, ..., k, denoted by g2 (U1 , ..., Uk ), is given by g2 (U1 , ..., Uk ) = {

k Z Y

−1

|Vj |

− p+1 2

Vj >Uj

j=1

−1

−1

−1

− 12

× φ1 (V1 2 U1 V1 2 , ..., Vk 2 Uk Vk × f (V1 , ..., Vk )dV1 ∧ ... ∧ dVk for

− 12

|I − Vj 2 Uj Vj Γp (αj )

|

}

)φ2 (V1 , ..., Vk )

p−1 2 .

5.4.2. Define Riemann-Liouville fractional integral of orders α1 , ..., αk of the second kind for many matrix variables as −α ,j=1,...,k

−α ,j=1,...,k

D2,Ujj ,j=1,...,k f = D2,(Uj, Bj ) k Z Y ={ j=1

f

Bj >Vj >Uj

× dV1 ∧ ... ∧ dVk .

|Vj − Uj |αj − Γp (αj )

p+1 2

f (V1 , ..., Vk )

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131

Specify φ1 and φ2 in Exercise 5.4.1 so that the M-convolution of the product g2 (U1 , ..., Uk ) agrees with this Riemann-Liouville fractional integral of orders αj , j = 1, ..., k of the second kind for many matrix variables, where Bj > O, j = 1, ..., k are constant real positive definite matrices. 5.4.3. Define Weyl fractional integral of orders α1 , ..., αk of the second kind for many matrix variables as −α ,j=1,...,k W2,Ujj,j=1,...,k f

={

k Z Y

j=1

Vj >Uj

|Vj − Uj |αj − Γp (αj )

p+1 2

}f (V1 , .., Vk )dV1 ∧ ... ∧ dVk .

Specify φ1 and φ2 in Exercise 5.4.1 so that g2 (U1 , ..., Uk ) agrees with this Weyl fractional integral in the many matrix variable case. 5.4.4. Specify φ1 and φ2 in Exercise 5.4.1 so that the M-convolution of the product g2 (U1 , ..., Uk ) agrees with the Kober fractional integral operator of −α ,j=1,...,k the second kind K2,Ujj ,βj ,j=1,...,k f defined in Section 5.4. 5.4.5. Evaluate the Kober fractional integral operator of the second kind in the equation (5.4.3) when f (V1 , ..., Vk ) = {

k Y

|Vj |−γj },

j=1

p−1 , j = 1, ..., k. 2

5.5. Fractional Integral Operators of the First Kind in the Case of Many Matrix Variables Let f1 (X1 , ..., Xk ) be a function of many p×p matrices and f2 (Y1 , ..., Yk ) be another function of another set of p × p matrices Y1 , ..., Yk . Let f1 be of the form k Y p+1 p+1 Γp (βj + αj ) { |Xj |βj − 2 |I − Xj |αj − 2 } Γ (αj )Γp (βj ) j=1 p

for O < Xj < I, 1 2

1 2

p−1 2 ,

p−1 2 ,j

(5.5.1)

= 1, ..., k. Let Uj =

Yj Xj−1 Yj , Vj = Yj , j = 1, ..., k. Then Uj , j = 1, ..., k can be called the symmetric ratio of matrices. The joint density of U1 , ..., Uk , when f1 and f2

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are statistical densities will be called the density of products. If f1 and f2 are not statistical densities then we are going to consider M-convolutions of ratios. In Section 5.4 we considered M-convolutions of products. The above 1 1 transformation is equivalent to Xj = V 2 Uj−1 V 2 , Yj = Vj , j = 1, ..., k. Let Z k Y 1 1 p+1 Γp (βj ) 1 |Vj2 Uj−1 Vj2 |βj − 2 { }g1 (U1 , ..., Uk ) = { Γ (β + αj ) Γ (αj ) Vj j=1 p j j=1 p k Y

1

1

p+1

× |I − Vj2 Uj−1 Vj2 |αj − 2 |Vj | × f (V1 , ..., Vk )dV1 ∧ ... ∧ dVk

p+1 2

−α ,j=1,...,k

= K1,Ujj ,βj ,j=1,...,k f (V1 , ..., Vk ).

|Uj |−(p+1) } (5.5.2) (5.5.3)

Then (5.5.3) will be taken as the definition of Kober fractional integral operator of the first kind of orders α1 , ..., αk with parameters β1 , ..., βk , in the many matrix variable case. Simplifying (5.5.2), we have a definition for generalized Kober fractional integral operator of the first kind in many matrix variables case. We note that −α ,j=1,...,k

K1,Ujj ,βj ,j=1,...,k f (V1 , ..., Vk ) Z k Y p+1 |Uj |−βj −αj |Vj |βj |Uj − Vj |αj − 2 } ={ Γp (αj ) Vj U j j j=1 1

1

1

1

× φ1 (V12 U1−1 V12 , ..., Vk2 Uk−1 Vk2 )φ2 (V1 , ..., Vk ) × f (V1 , ..., Vk )dV1 ∧ ... ∧ dVk for

p−1 2 .

5.5.2. Define Riemann-Liouville fractional integral of orders α1 , ..., αk of the first kind for many matrix variables as −α ,j=1,...,k

−α ,j=1,...,k

D1,Ujj ,j=1,...,k f = D1,(Aj j ,Uj ) k Z Y ={ j=1

f

Aj O −αj ,j=1,...,k K1,Uj ,βj ,j=1‘,...,k f (V1 , ..., Vk )dU1

p+1 2

∧ ... ∧ dUk .

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The Uj -integral is given by Z p+1 p+1 |Uj |sj − 2 |Uj |−βj −αj |Uj − Vj |αj − 2 dUj Uj >Vj

= |Vj |

−βj +sj − p+1 2

Z |Yj |αj −

p+1 2

Yj >O

× |I + Yj |

−(βj −sj + p+1 2 )

dYj , − 12

Tj = Uj − Vj , Yj = Vj = |Vj |

−βj +sj − p+1 2

− 12

Tj Vj

Γp (αj )Γp ( p+1 2 + β j − sj ) Γp ( p+1 2 + αj + βj − sj )

by evaluating the integral by using type-2 real matrix-variate beta integral, p−1 for