A04 07 0105

International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 4, Issue 7 (December 2014) PP: 01-05

Fractional Integration Operators of Certain Generalalized Hyper geometric Function of Two Variables Yashwant Singh1, Nanda Kulkarni2 (Research Scholar Shri JJTU) 1

2

Department of Mathematics, Government College, Kaladera, Jaipur, Rajasthan, India Department of Mathematics, Maharani Lakshmi Ammanni College for Women Bangalore Karnataka, India

Abstract: In the present paper the authors introduce two new fractional integration operators associated with

H -function of two variables. Three important properties of these operators are established which are the generalization of the results given earlier by several authors.

Key words: Fractional Integration Operators, H -function of two variables, Mellin Transform (2000 Mathematics subject classification : 33C99)

I. Introduction

H -function of two variables defined and represented by Singh and Mandia [10] in the following manner: , e j , E j ; R j  ,  e j , E j  o , n1: m2 , n2 : m3 , n2   a j , j ; Aj  , c j , j ; K j  , c j , j   H  x, y   H  xy   H p1 ,q1: p2 ,q2 ; p2 ,q2  xy b ,  ;B 1, p1, d ,  ,1,nd2 , ; L n2 1, p2 , f , F  1,,n3f , F ;S n31, p3  j j j 1,q j j 1,m j j j m 1,q j j 1,m j j j m 1,q 1 2 2 2 3 3 3   1   ,  2 ( )3 ( ) x y d d = (1.1) 2   1 4 L1 L2 The

Where n1

1  ,  

  1  a j 1

p1

  a

j  n1 1

2   

j

  j  Aj    1  b j   j  B j 

 1  c   c

j  n2 1

j

j

3   

  e

j  n3 1

Where

j

x and y are

  d

  j 

  j 

 1  e

(1.2)

j 1

n3

j 1 p3

  j  Aj  q1

n2

j 1 p2

j

j

Kj

j 1

  1  d q2

j  m2 1

j

  f

 E j 

 E j 

m2

Rj

j 1

  1  f

j

  j 



  j 

m3

q3

j  m3 1

j

j

(1.3) Lj

 Fj 



 Fj 

(1.4) Sj

not equal to zero (real or complex), and an empty product is interpreted as unity

pi , qi , ni , m j are non-negative integers such that 0  ni  pi , o  m j  q j (i  1, 2,3; j  2,3) . All the a j ( j  1, 2,..., p1 ), b j ( j  1, 2,..., q1 ), c j ( j  1, 2,..., p2 ), d j ( j  1, 2,..., q2 ), e j ( j  1, 2,..., p3 ), f j ( j  1, 2,..., q3 ) are complex parameters.

 j  0( j  1, 2,..., p2 ),  j  0( j  1, 2,..., q2 ) (not all zero simultaneously), similarly E j  0( j  1, 2,..., p3 ), Fj  0( j  1, 2,..., q3 ) (not all zero simultaneously). The exponents K j ( j  1, 2,..., n3 ), L j ( j  m2  1,..., q2 ), R j ( j  1, 2,..., n3 ), S j ( j  m3 1,..., q3 ) can take on nonnegative values. www.ijeijournal.com

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Fractional Integration Operators of Certain Generalalized Hyper geometric Function......

The contour

L1 is in  -plane and runs from i to i . The poles of   d j   j  ( j  1, 2,..., m2 ) lie to



the right and the poles of  1  c j   j



Kj

( j  1, 2,..., n2 ),  1  a j   j  Aj  ( j  1, 2,..., n1 ) to

the left of the contour. For K j ( j  1, 2,..., n2 ) not an integer, the poles of gamma functions of the numerator in (1.3) are converted to the branch points.

The contour

L2 is in  -plane and runs from i to i . The poles of   f j  Fj  ( j  1, 2,..., m3 ) lie to



the right and the poles of  1  e j  E j



Rj

( j  1, 2,..., n3 ),  1  a j   j  Aj  ( j  1, 2,..., n1 ) to

the left of the contour. For R j ( j  1, 2,..., n3 ) not an integer, the poles of gamma functions of the numerator in (1.4) are converted to the branch points.

The functions defined in (1.1) is an analytic function of p1

p2

q1

q2

j 1

j 1

j 1

j 1

p1

p3

q1

q3

j 1

j 1

j 1

j 1

x and y , if

U   j    j    j    j  0

(1.5)

V   Aj   E j   B j   Fj  0

(1.6)

The integral in (1.1) converges under the following set of conditions: n1

   j 

p1

m2

   

j 1

j  n1 1

n1

p1

   Aj  j 1



j  n1 1

j

j 1

m2

j 

Aj   Fj  j 1

q2

n2



 j Lj   j K j 

q2

Fj S j   E j R j 

j  m2 1



j  m2 1

p2

j 1

j  n2 1

n3

p3

j 1

q1

   

j  n2 1

0

(1.7)

E j   Bj  0

(1.8)

j

j 1

j

q1

j 1

1 1 (1.9) | arg x |  , | arg y |  2 2 The behavior of the H -function of two variables for small values of | z | follows as: H [ x, y]  0(| x | | y | ), max | x |,| y |  0

(1.10)

Where



 d j    2    j   For large value of | z | ,

  min  Re   1 j  m



 f j     Fj  

  min  Re   1 j  m 2



H [ x, y]  0| x | ' ,| y | '  , min | x |,| y |  0

(1.11)

(1.12)

Where

 c j 1   e j 1  ,  '  max Re  R j     1 j  n2 1 j  n3  j  E j    Provided that U  0 and V  0 .

 '  max Re  K j

(1.13)

If we take

K j  1( j  1, 2,..., n2 ), L j  1( j  m2  1,..., q2 ), R j  1( j  1, 2,..., n3 ), S j  1( j  m3 1,..., q3 ) in

H -function of two variables reduces to H -function of two variables due to [7].

If we set n1  p1  q1  0 , the H -function of two variables breaks up into a product of two H -function of (2.1), the

one variable namely 0,0:m2 , n2 :m3 , n3  H 0,0: p2 ,q2 : p3 ,q3  xy 

 1,n2 , c j , j n1, p2 : e j , E j ; R j 1,n3 , e j , E j n31, p3   : d j , j  , d j , j ; L j  : f j , F j  , f j , F j ; S j  1,m2 m2 1,q2 1,m3 m3 1,q3  : c j , j ; K j

www.ijeijournal.com

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Fractional Integration Operators of Certain Generalalized Hyper geometric Function...... m2 , n2

= H p2 , q2

  c j , j ; K j 1,n2 , c j , j n21, p2  m3 ,n3  x  d j , j 1,m2 , d j , j ;L j m21,q2  H p3 ,q3  

  e j , E j ;R j 1,n3 , e j , E j n31, p3   y  f j , Fj 1,m3 , f j , Fj ;S j m31,q3   

(1.14)

  0 , we then obtain , e j ,  E j ; R j  , e j ,  E j  0, n1: m2 , n2 :m3 , n3    a j , j ; A j  , c j ,  j ; K j  , c j , j    2 H p1 ,q1: p2 ,q2 : p3 ,q3  xy b , ;B  1, p1, d ,  ,1,nd2 , ;L n21, p2 , f , F  1,,n3f , F ;S n31, p3  j j j 1,q j j 1,m j j j m 1,q j j 1,m j j j m 1,q 1 2 2 2 3 3 3   , e j , E j ; R j  , e j , E j  0, n1: m2 , n2 :m3 , n3  x  a j , j ; A j  , c j , j ; K j  , c j , j  1, p 1,n2 n2 1, p2 1,n3 n3 1, p3  = H p1 ,q1: p2 ,q2 : p3 , q3  y b ,  ; B 1, d ,  , d ,  ; L , f , F , f , F ; S            j j j 1,q1 j j 1,m2 j j j m2 1,q2 j j 1,m3 j j j m31,q3  , e j , E j ; R j  , e j , E j  0, n1: m2 , n2 :m3 , n3  1  a j , j ; A j  , c j , j ; K j  , c j , j   H p1 ,q1: p2 ,q2 : p3 ,q3  1x b ,  ; B 1, p1, d ,  ,1,nd2 , ; L n21, p2 , f , F  1,,n3f , F ;S n31, p3   y j j j 1,q1 j j 1,m2 j j j m2 1,q2 j j 1,m3 j j j m31,q3  ,1 d j , j ; L j  ,1 f j , Fj  ,1 f j , F j ; S j  0, n1: m2 , n2 :m3 , n3  x 1b j ,  j ; B j  ,1 d j , j  1,q 1,m m 1,q 1,m m 1,q  = H p1 ,q1: p2 ,q2 : p3 ,q3  y 1 a , ; A 1 , 1c , ; K 2 , 1c , 2 2 , 1e , E ; R 3 , 1e , E 3 3    j j j 1, p1  j j j 1,n2  j j n21, p2  j j j 1,n3  j j n31, p3  If

(1.15)

(1.16)

II. Definitions We introduce the fractional integration operators by means of the following integral equations: x

Y   ,, ;;rx f ( x)   rx  r  1  t   x r  t r  f (t ) H  UU  dt 

(2.1)

0

And

N 

 ,  ;r  , ;x



f ( x)   rx

x

t

  r  1

t

r

 x r  f (t ) H  VV  dt 

(2.2)

0

Where U and V represent the expressions m

n

m

n

 tr   tr   xr   xr  1  and  r   r  1  r  r  x   x  t   t  Respectively, r , m, n are positive integers and 1 1 | arg  |  , | arg  |  .

2 2 The conditions of the validity of these operators are as follow: (i) 1 

p, q  ,

1 1  1 p q

 dj fj  1 Re    rm  rm    ,  j Fj  q   dj fj  1 Re    rn  rn    ,   j Fj  q   dj fj  1 Re    rm  rm    ,   j Fj  p  (iii) f ( x)  Lp (0, ). (ii)

The last condition ensures that If we set n1

Y and N both exists and also that both belongs to Lp (0, ) .

 p1  q1  0 , we obtain the operators involving the product of two H -functions.

If we take

K j  1( j  1, 2,..., n2 ), L j  1( j  m2  1,..., q2 ), R j  1( j  1, 2,..., n3 ), S j  1( j  m3 1,..., q3 ) we obtain the operators involving

H -function of two variables. www.ijeijournal.com

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Fractional Integration Operators of Certain Generalalized Hyper geometric Function......

III. Mellin Transform The Mellin transform of

f (t ) will be defined by M [ f (t )] . We write s  p 1  it where p and t are real. If

p  1, f (t )  Lp (0, ) , then 

p  1, M [ f (t )]   t s 1 f (t )dt

(3.1)

0 x

p  1, M [ f (t )]  l.i.m.  t s 1 f (t )dt

(3.2)

1/ x

Where l.i.m. denotes the usual limit in the mean for L p -spaces.

Theorem 1. If

1 1 f ( x)  Lp (0, ),1  p  2[or f ( x)  M p (0, )and p  2] | arg  |  ,| arg  |  , (, )  0 2 2   dj fj  dj fj  1 1 Re    rm  rm    , Re    rn  rn    , then     j Fj  q j Fj  q  





o , n1:

m2 , n2 : m3 , n2

M Y   ,, ;;rx f ( x)   H p1  2,q1  2: p2 ,q2 ; p2 ,q2   a j , j ; Aj 1, p ,   s ,m , c j , j ; K j 1,n , c j , j n 1, p , e j , E j ; R j 1,n , e j , E j n 1, p  r 1  2 2 2 3 3 3   xy  M [ f ( x)] (3.3) 1  s    b j ,  j ; B j 1,q1 ,    r ,m n , d j , j 1,m2 , d j , j ; L j m2 1,q2 , f j , Fj 1,m3 , f j , Fj ;S j m31,q3  Where M p (0, ) denotes the class of all functions f ( x) of Lp (0, ) with p  2 which are inverse Mellin transforms of functions of Lq (, ) .

Proof: From (2.1), it follows that



M Y 

 ,  ;r  , ;x





f ( x)    x rx s 1

  r  1

0 

=

t



x

x

 t x

r

 t r  f (t ) H  UU  dtdx 

0

f (t )dt  x s  r  2  x r  t r  f (t ) H  UU  dx ,

0



0

The change of order of integration is permissible by virtue of de la Vallee Poisson’s theorem ([2],p.504)under the conditions stated with the theorems. The theorem readily follows on evaluating the x -integral by means of the formula , c j , j ; K j  , c j , j  , e j , E j ; R j  , e j , E j  o , n1: m2 , n2 : m3 , n2  k l  a j , j ; A j   (1  x)  1 H p1 ,q1: p2 ,q2 ; p2 ,q2  xxk (1(1xx))l b ,  ; B 1, p1, d ,  ,1,nd2 , ; L n21, p2 , f , F  1,,n3f , F ;S n31, p3  dx j j j 1,q j j 1,m j j j m 1,q j j 1,m j j j m 1,q 1 2 2 2 3 3 3   0 , e j , E j ; R j  , e j , E j  o , n1: m2 , n2 : m3 , n2    a j , j ; A j  ,(1 , k ), c j , j ; K j  , c j , j   1, p 1,n n 1, p 1,n3 n3 1, p3 = H p1  2,q1  2: p2 ,q2 ; p2 ,q2   b ,  ; B 1,(1   , k l ), d , 2 , d , 2; L 2 (3.4)   j j j 1,q j j 1,m  j j j m 1,q , f j , F j 1,m ,  f j , F j ; S j m 1,q  1 2 2 2 3 3 3   1

x

 1

Where

  dj fj  dj fj  1 1 Re    k  k   0, Re    k  k   0 | arg  |  ,| arg  |  , (, )  0 ,   j Fj  j Fj  2 2   which follows from the definition of the H -function of two variables and Beta function formula.

In a similar manner, the following theorems can be establisded.

Theorem 2. If

1 1 f ( x)  Lp (0, ),1  p  2[or f ( x)  M p (0, )and p  2] | arg  |  ,| arg  |  ,(, )  0 2 2   dj fj  dj fj  1 1 1 1 Re    rn  rn    , Re     r  rm  rm    ,   1 then   j Fj  q j Fj  q p q   www.ijeijournal.com

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Fractional Integration Operators of Certain Generalalized Hyper geometric Function......





o , n1:

m2 , n2 : m3 , n2

M N   ,, ;;xr f ( x)   H p1  2,q1  2: p2 ,q2 ; p2 ,q2   a j , j ; Aj 1, p , 1  s ,m , c j , j ; K j 1,n , c j , j n 1, p , e j , E j ; R j 1,n ,e j , E j n 1, p  r 1  2 2 2 3 3 3  (3.5)    M [ f ( x)]  s    b j ,  j ; B j 1,q1 ,    r ,m n , d j , j 1,m2 , d j , j ; L j m2 1,q2 , f j , Fj 1,m3 , f j , Fj ;S j m31,q3  1 1 Theorem 3. If f ( x)  Lp (0, ), g ( x)  Lp (0, ),| arg  |  ,| arg  |  , (, )  0 , 2 2   dj fj  dj fj   1 1 1 1 Re    rm  rm   max  ,  , Re   r  rm  rm   0,   1 then     j Fj  j Fj  p q  p q   



0

0

 ,  ;r  ,  ;r  g ( x)Y   , ;x f ( x) dx  f ( x) N   , ;x g ( x) dx

(3.6)

IV. Formal Properties of the Operators Here, we give some formal properties of the operators which follow as consequences of the definitions (2.1) and (2.2).

x 1Y   ,, ;;rx1 f ( x 1 )   N   ,, ;;rx f ( x)    x 1 N   ,, ;;xr1 f ( x 1 )   Y   ,, ;;xr f ( x)      ,  ;r x Y   , ; x f ( x)   Y   ,;,x ;r x f ( x) 

(4.1) (4.2) (4.3)

x N   ,, ;;xr f ( x)   N   ,;,x ;r x f ( x) 

(4.4)

If

Y   ,, ;;rx f ( x)   g ( x) , then Y   ,, ;;rx f (cx)   g (cx)

(4.5)

And if

N   ,, ;;xr f ( x)    ( x) , then N   ,, ;;xr f (cx)    (cx)

(4.6)

In conclusion, it is interesting to observe that double integral operators associated with H -function of two variables can be defined in the same way and their various properties, analogous to the integral operators studied in this paper can be obtained.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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Bromwich, T.J.T.A.; Theory of Infinite Series, Macmillan, London (1954).

Erdelyi, A.; On fractional integration and application to the theory of Hankel transforms, Quar. J. Math., Oxford Series 2(44), (1940),293-303.

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Kalla, S.L. and Saxena, R.K.; Integral operators involving hypergeometric functions, Math. Zeitscher, 108, (1969), 231-234.

Lowndes, J.S.; A generalization of the Erdelyi-Kober operators, Proc. Edinburgh Math. Soc. 17(2), (1970), 139-148.

Mittal, P.K. & Gupta, K.C.; An integral involving generalized function of two variables. Proc. Indian Acad. Sci. Sect. A( 75)1961), 67-73.

Munot, P.C. and Kalla, S.L.; On an extension of generalized function of two variables, Universidad Nac. De. Tuc. Rev. Ser. A(1971),21-27.

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Singh,Y. and Mandia, H.; A study of H -function of two variables, International Journal of Innovative research in science,engineering and technology,Vol.2,(9)(2013),4914-4921 Saxena, R.K.; On fractional integration operators, Math. Zeitcher, 96(1967), 288-291.

12. Saxena, R.K.; Integrals of products of H -functions, Universidad Nac. De. Tuc. Rev. Ser. A(1971),185-191.

13. Saxena, R.K. and Kumbhat, R.K.; A generalization of Kober operators, Vijnana Parishad Anusandhan Patrika, India, 16(1973), 31-36.

Saxena, R.K. and Kumbhat, R.K.; Fractional Integration operators of two variables, The Proc. Of the Indian Academy of Sci., 78 Sec. A (1973),177-186.

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