Kampe De Feriet’s Function and Lauricella Function of Matrix Arguments in Complex Case

Kampe De Feriet’s Function and Lauricella Function of Matrix Arguments in Complex Case

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

ETRASCT' 14 Conference Proceedings

Dr. Sandeep Mathur1

Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan)

E-mail :- mathur.sandeep1979@gmail.com

Dr. Anjali Mathur2

Associate Professor, Department of Mathematics, Jodhpur Institute of Engg. & Technology, Jodhpur (Rajasthan)

E-mail :- anjali24mathur@gmail.com

ABSTRACT

In this paper, we have studied the Mathais definitions of the Kampé de Fériets functions, the Lauricella functions of matrix arguments in complex case.

1. INTRODUCTION

   Techniques of Mathai [1,2] for positive symmetric matrix, we define Kampé de Fériets, Lauricella and other functions of matrix argument in complex case. All matrices used in this paper are hermitian positive definite. All the matrices appearing in this paper are p x p real Hermitian positive definite and meanings of all the other symbols used are the same as in the work of Mathai [1,2]

   FUNCTION OF MATRIX ARGUMENT IN THE COMPLEX CASE :

   ~

   We consider real valued scalar function of a single matrix argument of the type Z =

   ~ ~ ~ ~

   X + i Y where X and Y are p x p matrices with real elements and i

   1 as well as

   scaler functions of many matrices

   ~

   Zj , j = 1, 2, .K where each

   ~ ~

   Zj is of the type Z

   above in the real case. We confined our discussion to the situation where the argument matrix was real symmetric positive definite. This was done so that the fractional power of matrices and functions of such matrices could be uniquely defined. Corresponding properties are of we restrict to the class of Hermitian positive definite matrices.

   Definition : Hermitian positive definite matrix due to Mathai [3], We will denote the

   ~ ~ ~ ~ ~

   conjugate of Z by Z if Z hermitian, then Z = Z*, that is

   ~ ~ ~ ~ ~ ~ ~ ~

   Z = Z* X + i Y = ( X + i )* = X + i Y

   ~ ~ ~ ~

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   X = X and Y = Y

   ~ ~

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   ~

   Thus X is the symmetric and Y is skew symmetric. Further if

   ~

   Z is hermitian positive

   definite, then all the eigen values of Z are real and positive. Further, matrix variate gamma in the complex case is

   p

   ~ ()

   p ( p1) ( )( 1)……. ( 1)

   = 2

   ~

   We will use the notation

   Z 0

   to indicate that

   ~

   Z is hermitian positive definite.

   Constant matrices will be written without a tilde whether the elements are real or complex unless it has to be emphasized that the matrix involved has complex elements. Then in that case a constant matrix will also be written with a tilde.

2. DEFINITIONS

   1. The Kampé de Fériets function

      a :

      r

      b ;

      q

      c ; ~ ~

      k

      F

      s:m;n

      r:q; k

      s

      m

      n

      s:m; n

      Fr:q;k

      : ;

      ;

      X,Y

      of matrix arguments is defined as that class of functions for which the M-transform is the following:

      MFr:q;k

      ~ 1 p ~ 2 p

      s:m;n

      X Y

      ~ ~

      a :

      r

      X 0 Y 0

      b ; c ; ~

      q

      k

      ~ ~ ~

      s:m; n

      s

      m

      n

      Fr:q; k

      : ;

      ;

      X,Y

      dXdY

      s ~

      m

      ~

      n

      ~

      r

      ~ a

      p

      j1

      j p

      j1

      j p

      j1

      j p j 1 2

      j1

      r ~ a

      q

      ~ b

      k

      ~ c

      s

      ~

      p

      j1

      j p

      j1

      j p

      j1

      j p j 1 2

      j1

      q ~ b

      k ~ c

      ~ ~

      p j 1 p j 2 p 1 p 2

      j1 j1

      (2.1)

      m ~ n ~

      p j

      j1

      1 p j 2

      j1

      1 2 j 1 2 j 1 2 j 1

      for Re , ,a , j 1,…,r; , j 1,…,s; b , j 1,…,q;

      j 1 , j 1,…, m;c j

      2

      , j 1,…, k; j

      2

      , j 1,…,n p 1

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      ISSN: 2278-0181

      ETRASCT' 14 Conference Proceedings

   2. The Lauricella function

      Fn Fn a, b ,…, b

      ;c ,…,c

      ; ~ ~

      A A 1

      n 1 n

      X1 ,…, Xn

      of matrix arguments is defined as that class of functions which has the following matrix transform:

      A

      MFn

      …

      ~ 1 p

      X1

      ~

      …Xn

      n p

      ~

      ~

      X1 0 X n 0

      Fn a, b ,…, b

      ;c ,…,c

      ; ~

      ~ ~ ~

      A 1 n 1

      n X1 ,…,

      Xn dX1…dXn

      n ~ c

      n

      ~ b

      ~

      a

      …

      ~

      p

      j1

      j p j

      j1

      j p 1

      n p j

      n

      j1

      (2.2)

      ~ a n

      ~ b

      n

      ~ c

      p p

      j1

      j p j j

      j1

      j j j j j 1 n

      for Reb , c , , a … p 1; j 1,…,n.

   3. Fn Fn a

      ,…,a

      , b ,…,b

      ;c; ~

      ~

      B B 1 n 1

      n X1 ,…, Xn

      B

      MFn

      …

      ~ 1 p

      X1

      ~

      …Xn

      n p

      ~

      ~

      X1 0 X n 0

      Fn a

      ,…,a , b ,…, b

      ;c; ~

      ~ ~ ~

      1. 1 n 1

         n X1 ,…,

         Xn dX1…dXn

         ~ c n ~ a

         ~ b

         ~ a

         p p j j p j j p j

         p

         p

         j

         j1

         (2.3)

         n

         p

         j

         1

         n

         j1

         ~

         a ~

         b ~

         c

         …

         j j j j j 1 n

         for Rea , b , , c … p 1; j 1,…,n.

   4. n

      F

      C

      Fn a, b, c

      ,…, c n

      ~ ~

      ;

      X1 ,…, X n

      C

      C

      1

      MFn

      …

      ~ 1 p

      X1

      ~

      …Xn

      n p

      ~

      ~

      X1 0 X n 0

      Fn a, b, c ,…,c

      ; ~

      ~ ~ ~

      1. 1 n

         X1 ,…,

         Xn dX1…dXn

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         n ~

         c n ~ a

         …

         ~ b

         …

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         p

         j p 1

         n p 1

         n n ~

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         j1 j1

         (2.4)

         ~ ~ n ~

         p j

         a b c

         j1

         p p j1 p j j

         j j j 1 n 1 n

         for Rec , , a … , b … p 1; j 1,…, n.

   5. Fn Fn a, b ,…,b

      ;c; ~ ~

      1. D 1

      n X1 ,…, Xn

      D

      MFn

      …

      ~ 1 p

      X1

      ~

      …Xn

      n p

      ~

      ~

      X1 0

      Fn a, b ,…, b

      X n 0

      j

      p

      1

      ;c; ~

      ~ ~ ~

      p

      D 1 n

      X1 ,…,

      Xn dX1…dXn

      n

      ~

      p c

      ~

      j1

      ~ b

      • j

      ~

      ~

      a

      … n

      (2.5)

      n

      p

      j

      p

      p a j1

      ~ b

      ~

      c

      … n

      p

      j

      1

      for Reb

      , , a … ,c …

      p 1; j 1,…,n.

      j j j 1 n 1 n

3. KAMPÉ DE FÉRIETS FUNCTION OF MATRIX ARGUMENTS

   Theorem 3.1

   : , ;',';

   ~ ~

   ~ ~ '

   p

   p

   F1:2; 2

   X,Y ~ ~ ~ ~

   1:1;1

   : ; ';

   ' ''

   p p p p

   I I ~ p ~ ' p

   ~ p ~ ' ' p F

   ~ 12 ~ ~ 12

   ~ 12 ~ ~ 12 ~

   U V I

   0 0

   U I V

   1 , ,

   '; ; U XU

   , V YV dUdV

   for Re , ', ,'' p 1

   …(3.1)

   Proof: From definition (2.1) we deduce that,

   1:2; 2

   1:1;1

   ~ 1 p

   M(F )

   X

   ~ 2 p

   Y

   ~ ~

   F1:2;2 : ,;',';

   X0 Y0

   ~ ~ ~ ~

   1:1;1

   : ; ';

   X,YdXdY

   ~ ~ ~ '~ ~ ~

   p

   p

   p

   p

   1

   2

   p

   1

   ~ p ~ ~ ~ ~ ~ ~ 1

   ' '

   p p p p p p 1 2 p 1

   p

   2

   ~ '

   ~

   ~ '

   ~ ~

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   p

   2

   ' p

   2 p 1 p 2

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   for Re , , , , ,' , ' , , ,'

   p 1

   1 2 1 2 1 1 2 2 1 2 1 2

   Now taking the M-transform of the right side of eq.(3.1) with respect to the variables

   ~ ~

   1

   2

   X, Y and the parameters ,

   respectively, we have,

   ~ 1 p ~ 2 p F ,,'; ; ~ 12 ~ ~ 12

   ~ 12 ~ ~ 12 ~ ~

   (3.3)

   X Y 1

   ~ ~

   U XU

   , V YV

   dXdY

   X0 Y0

   Applying the transformations,

   ~ ~ 12 ~ ~ 12 ~ ~ 12 ~ ~ 12

   1

   1. U XU , Y V XV

      1

      (implying thereby

      ~ ~ p ~ ~ ~ p ~

      ~ ~ ~ ~ ~ ~

      ) in the

      dX1 U dX,dY1 V

      dY, and X1 U X ,

      Y1 V Y

      expression (3.3) and then making use of Mathais definition of M-transform of an Appells function F1 we get,

      ~ ~ ~ ~

      p 1 p 2

      ~ 1 ~ 2 p p 1 p 2 p 1 2 ~ ~

      U V ~ ~ ~ '~

      '

      (3.4)

      p p p p 1 2

      Substituting this expression on the right side of eq. (3.1) and integrating out the variables

      ~ ~

      U and V in the resulting expression by using a type-1 Beta integral we finally obtain

      1:1;1

      MF1:2;2 as given by eq. (3.2)

4. LAURICELLA FUNCTIONS OF MATRIX ARGUMENTS

Theorem 4.1

Fn Fn , ,…,

; ,…,

; ~ ~

A A 1

n 1 n

X1,…, Xn

~ 1

e

~ ~ n p n , ,…,

T

; ,…,

; ~

~ ~ 12 ~

~ 12 ~

p

~

n

T0

tr T

A

1 n1 1

n X1

,…,

Xn1 , T

Xn T

dT

(4.1)

n

for Re p 1

Proof: Taking the M-transform of the right side of eq. (4.1) with respect to the variables

~ ~

X1 ,…,Xn and the parameters 1,…,n

respectively, we get

~ 1p

~ n p

n

~ ~ ~ 12 ~ ~ 12 ~

~ ISSN: 2278-0181

1 n

…

~

X1

…Xn

A ,1 ,…,n1; 1 n ;X1,…,Xn1 ,T XT

dX …dX

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X10

Xn 0

(4.2)

which on applying the transformation

~ ~ 12 ~ ~ 12

~ ~ p ~

~ ~ ~

n

1. T XT

(with dYn T dXn

and Yn T Xn )

~ n1 ~

p

n ~

~ n p 1 … n i1

i p j p j

i

j1

T ~

n 1 ~

n ~

(4.3)

p

i1 p

i j1 p j j

Substituting this expression on the right side of eq. (4.1) and then integrating out

~

T in the

A

resulting expression by using a Gamma integral produces MFn as given by eq. (2.2).

Theorem 4.3

Fn ,…, , ,…,

; ; ~ ~

1. 1 n 1

   n X1,…, Xn

   ~ 1

   e

   ~ ~ n p n ,…,

   T

   , ,…,

   ; ; ~

   ~ ~ 12 ~

   ~ 12 ~

   p

   ~

   n

   T0

   tr T

   1 1

   n 1 n1

   X1 ,…,

   Xn1 , T

   Xn T dT

   n

   for Re p 1

   (4.4)

   Theorem 4.5

   Fn ,; ,…,

   ; ~ ~

2. 1 n

   X1,…, Xn

   ~ 1

   e ~ ~ p n ; ,…,

   T

   ;~ 12 ~

   ~ 12

   ~ 12 ~

   ~ 12 ~

   (4.5)

   p

   T0

   tr T

   ~

   2

   1 n T

   X1T

   ,…, T

   Xn T dT

   for Re p 1

   Theorem 4.6

   Fn , ,…,

   ; ; ~ ~

3. 1 n

X1 ,…, Xn

~ 1

e

~ ~ n p n , ,…,

; ; ~

~ ~ 12 ~

~ 12 ~

(4.6)

p

~

n

T0

tr T T

D 1 n1

X1 ,…,

Xn1 , T

Xn T dT

for Ren p 1

REFERENCES

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

ETRASCT' 14 Conference Proceedings

1. Mathai A. M. (1992) : Jacobians of Matrix Transformations-I, Center for Mathematical Sciences, Trivandrum, India.

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