APPELL’S AND HUMBERT’S DISTRIBUTIONS OF MATRIX VARIATE IN THE COMPLEX CASE

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APPELL’S AND HUMBERT’S DISTRIBUTIONS OF MATRIX VARIATE

IN THE COMPLEX CASE

1

Ms. Pooja Singh, 2Dr. Rajesh Prasad and 3 Prof. (Dr.) Harish Singh 1

Research Scholar, NIMS University, Jaipur, Rajasthan, India

2

Assistant Professor, Department of Mathematics, NIET, NIMS University, Jaipur, Rajasthan,India

3

Professor, Department of Business Administration, Maharaja Surajmal Institute, Affiliated to Guru Gobind Singh Indraprastha University, New Delhi, India

ABSTRACT

The aim of this paper is to investigate matrix variate generalizations of multivariate

Appell’s and Humbert’s families of distributions in the complex case. The multivatiate Appell’s and Humbert’s families of distributions have been proposed and studied recently

by Mathai and also by Saxana,Sethi and Gupta. Many known or new results have been

made with the help of Appell’s functions of matrix arguments-one each for the function

1

F and F and one for the function 3 F in complex case. 2

INTRODUCTION

Appell’s functions of matrix arguments have earlier been studied by Mathai [4, 5, 6] and also by Saxena, Sethi and Gupta [7]. In the present paper we have utilized Mathai’s definitions for all the

functions studied. Upadhyaya and Dhani have given integral representation associated with

Appell’s Humbert’s functions of matrix arguments in the case of real symmetric positive

definite. We have given in this paper further generalization of these results in the case of

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paper are (pxp) real Hermitian positive definite matrices and the meanings of all the other

symbols used are the same as in the works of Mathai [3, 4].

1. Preliminary Definitions

Definition 1.1: The Appell’s function F = 1 F (a, b, b1 ; c; – X–Y) of matrix argument is

defined as that function for which the M-transform (Matrix-transform) is the following:

M (F ) = 1



X0



Y0 Xρ₁p Yρ₂pF1(a, b, b; c; – X– Y) dXdY

= ) ρ ρ c ( ~ ) b ( ~ ) b ( ~ ) a ( ~ ) ρ b ( ~ ) ρ b ( ~ ) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ 2 1 p p p p 2 p 1 p 2 1 p 2 p 1 p p                 

…. (1.1)

for Re (a – 1 – 2, b – 1, b – 2, c – 1– 2, 1, 2) > p-1

Definition 1.2: F = 2 F (a, b, b2 ; c; – X–Y)

M (F ) = 2



X0



Y0 Xρ₁p Yρ₂pF2(a, b, b; c, c; – X– Y) dXdY

= ) ρ c ( ~ ) ρ c ( ~ ) b ( ~ ) b ( ~ ) a ( ~ ) ρ b ( ~ ) ρ b ( ~ ) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ ) c ( ~ 2 p 1 p p p p 2 p 1 p 2 1 p 2 p 1 p p p                     

…. (1.2)

for Re (a – 1 – 2, b – 1, b – 2, c – 1, c – 2, 1, 2) > p-1

Definition 1.3: F = 3 F (a, a3 , b, b; c; – X–Y)

M (F ) = 3



X0



Y0 Xρ₁p Yρ₂pF3(a, a b, b; c; – X– Y) dXdY

= ) ρ ρ c ( ~ ) b ( ~ ) b ( ~ ) a ( ~ ) a ( ~ ) ρ b ( ~ ) ρ b ( ~ ) ρ a ( ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ 2 1 p p p p p 2 p 1 p 2 1 p 2 p 1 p p                     

…. (1.3)

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Definition 1.4: F = 4 F (a, b; c, c4 ; – X – Y )

M (F ) = 4



X0



Y0 Xρ₁p Yρ₂pF4(a, b; c, c; – X – Y ) d X d Y

= ) ρ c ( ~ ) ρ c ( ~ ) b ( ~ ) a ( ~ ) ρ ρ b ( ~ ) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ ) c ( ~ 2 p 1 p p p 2 1 p 2 1 p 2 p 1 p p p                  

…. (1.4)

for Re (a – 1 – 2, b – 1 – 2, c – 1 – 2, 1, 2) > p-1

Definition 1.5: The Humbert’s function 1 = 1 (a, b; c; – X – Y ) of matrix arguments is

defined as that function which has the following M-transform:

M (1) =



X0



Y0 Xρ₁p Yρ₂p 1(a, b; c; – X ,– Y ) d X d Y

= ) ρ ρ c ( ~ ) b ( ~ ) a ( ~ ) ρ b ( ~ ) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ 2 1 p p p 1 p 2 1 p 2 p 1 p p             

…. (1.5)

for Re (a – 1 – 2, b – 1, c – 1 – 2, 1, 2) > p-1

Definition 1.6: 1 = 1 (a, b; c; – X – Y )

M (1) =



X0



Y0 Xρ₁p Yρ₂p-1 1(a, b; c, c; – X ,– Y ) d X d Y

= ) ρ c ( ~ ) ρ c ( ~ ) b ( ~ ) a ( ~ ) ρ b ( ~ ) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ ) c ( ~ 2 p 1 p p p 1 p 2 1 p 2 p 1 p p p                 

…. (1.6)

for Re (a – 1 – 2, b – 1, c – 1 c – 2, 1, 2) > p-1

Definition 1.7: 2 = 2 (a; c, c; – X – Y )

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=

) ρ c ( ~ ) ρ c ( ~ ) b ( ~ ) a ( ~

) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~ ) c ( ~

2 p

1 p p p

2 1 p 2 p 1 p p p

      

       

…. (1.7)

for Re (a – 1 – 2, c – 1, c – 2, 1, 2) > p-1

Definition 1.8: 2 = 2 (a; b; c; – X – Y )

M (2) =



X0



Y0 Xρ₁p Yρ₂p-1 2(a, b; c; – X ,– Y ) d X d Y

=

) ρ ρ c ( ~ ) b ( ~ ) a ( ~

) ρ b ( ~ ) ρ a ( ~ ) ρ ( ~ ) ρ ( ~ ) c ( ~

2 1 p p p

1 p 1 p 2 p 1 p p

    

      

…. (1.8)

for Re (a – 1, b – 1, c – 2, 1, 2) > p-1

2. Appell’s Functions of Matrix Arguments. THEOREM 2.1:

1

F (, , ; c; – X – Y )

=

) β β ( ~ ) β ( ~ ) β ( ~

) γ ( ~

p p p

p

      







I 0



I 0

p -β β

| U

|  |V|β-p|IU|γ-ββ-p |IV|β-p

 

 

 

 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

) V (I ) U Y U V ) V (I U X U ) V (I

I dU d V ... (2.1)

for 0 < U < I, 0 < V < I and for Re (, ,  –  – ) > p – 1

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0

X



Y0 p

1

ρ

X  Yρ₂p-1

 

 



 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

) V ) U Y U V ) V (I U X U ) V (I

I d

X d Y ... (2.2)

Making use of the transformations,

1

X = (I V )1/2X U1/2, Y = V1 1/2Y U1/2V1/2 with, dX = 1 |IV|p|U|pdX , dY = | V |1 p |U |p

dY and |X | = |I – V | | U | |1 X |, |Y | = | V | | U | |1 Y |

in the above expression and then integration out the variables X and 1 Y by using a type – 2 1

Dirichlet integral we get,

|U |ρ1ρ2_|_{V |}ρ2_{|I –}_{V |}ρ1

) ( ~

) ρ ρ a ( ~ ) ρ ( ~ ) ρ ( ~

p

2 1 p 2 p 1 p

α



    

… (2.3)

Substituting this expression on the right side of eq. (2.1) and then integrating out the variablesU

and V in the resulting expression by using a type – 1 Beta integral generates M(F ) as give by 1

eq. (1.1)

THEOREM 2.2:

|P | β F2(, , ; ,; – X , – P–1/2Y – P–1/2)

= ) β ( Γ ~ 1

p 

P β ) T P tr(

| T | 0e

T  



 1(, ; ,; – X – T1/2Y T1/2) …(2.4)

for Re () > – 2.

PROOF: Taking the M-transform of the right side of eq. (2.4) with respect to the variables X and Y and the parameters 1 and 1 respectively, we get,

0 X







Y0 p

1

ρ

X  Yρ2p

 1

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and then the use of definition (1.6) leads us to,

|T |ρ2

) ρ γ ( ~ ) ρ γ ( ~ ) β ( ~ ) α ( ~

) ρ β ( ~ ) ρ ρ α ( ~ ) ρ ( ~ ) ρ ( ~ ) γ ( ~ ) γ ( ~

2 p

1 p p p

1 p 2 1 p 2 p 1 p p p

      

         

… (2.6)

Substituting this expression the right side of eq. (2.4) and then integrating out T in the resulting

expression by using a Gamma integral gives,

|P |βρ2

) ρ γ ( ~ ) ρ γ ( ~ ) β ( ~ ) α ( ~ ) β ( ~

) ρ β ( ~ ) ρ ρ α ( ~ ) ρ ( ~ ) ρ ( ~ ) γ ( ~ ) γ ( ~ ) ρ β ( ~

2 p

1 p p p p

1 p 2 1 p 2 p 1 p p p 1 p

        

            

… (2.7)

Now, Taking the M-transform of the left side of eq. (2.4) with respect to the variables X and Y

and the parameters 1 and 1 respectively, we get,

0 X







Y0 p

1

ρ

X  p

2

ρ

Y  |P |βF (2 ,,;,;–X – P-1/2Y P-1/2)dX d Y …(2.8)

and then using the definition (1.2) yields the same result as in eq. (2.7) above.

THEOREM 2.3:

|F | (3 ,, , ; ,; – X ,– Y )

=

) γ ( Γ~ ) ( Γ ~

) γ (γ Γ~

p p

p

  

γ



 I 0

P γ

| U

| IUγP ₂F (1 , ; ; –U1/2X U1/2)

1

2F (, ; ; – (I –U

1/2

Y (I – U1/2)dU …(2.9)

for Re 0 < U < I and for re (, ) > p– 1.

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0 X







Y0 p

1

ρ

X  p

2

ρ

Y  ₂F (1 , ; ; –U1/2X U1/2) 

1

2F (, ; ; – (I –U

1/2

Y (I – U1/2) dX d Y …(2.10)

Applying the transformations,

1

X U1/2 = X U1/2) Y = (I – U )1 1/2Y (I – U1/2 with dX = | U |1 p dX ,

dY = (I – U )1 p dY and |X | = | U | |1 X |, |Y | = |I – U | |1 Y |

to the above expression and then applying eq. (2.3.5) page 38 of Mathai [4] leads us to,

|U |ρ1_{|I –}_{U |}ρ1

) α ( ~ ) ρ γ ( ~ ) β ( ~ ) α ( ~

) γ ( ~ ) ρ β ( ~ ) ρ α ( ~ ) γ ( ~

p 1 p p p

p 2 p

1 p

p

    

       



) ρ γ ( ~ ) β ( ~

) ρ β ( ~ ) ρ α ( ~ ) ρ ( ~ ) ρ ( ~

2 p

p

1 p 1 p 2 p 1 p

    

     

… (2.11)

Substituting this expression on the right side of eq. (2.9) and then integration out the variable U

in the resulting expression by using a type – 1 Beta integral results in M(F ) (definition (1.3)). 3

References

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3. Mathai A.M. (1992). Jacobians of Matrix Transformations L Centre for Mathematical

Sciences, Trivandrum, India.

4. Mathai A.M. (1993). Hypergeometric Functions of Several Matrix .Arguments. Centre

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Page 16

5. Mathai A.M. (1993). Appell's and Humbert's Functions of Matrix Arguments, Linear

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