A04 07 0105 by IJEI Journal

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

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

  a

j  n1 1

2   

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

 1  c   c

j  n2 1

3   

  e

j  n3 1

Where

x and y are

  d

  j 

  j 

 1  e

(1.2)

j 1

j 1 p3

  j  Aj  q1

j 1 p2

j 1

  1  d q2

j  m2 1

  f

 E j 

 E j 

j 1

  1  f

  j 



  j 

j  m3 1

(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



( 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



( 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

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 

   

j 1

j  n1 1

   Aj  j 1



j  n1 1

j 1

j 

Aj   Fj  j 1



 j Lj   j K j 

Fj S j   E j R j 

j  m2 1



j  m2 1

j 1

j  n2 1

j 1

   

j  n2 1

0

(1.7)

E j   Bj  0

(1.8)

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)

And

N 

 ,  ;r  , ;x



f ( x)   rx

t

  r  1

t

 x r  f (t ) H  VV  dt 

(2.2)

Where U and V represent the expressions m

 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



 t x

 t r  f (t ) H  UU  dtdx 

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



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   



 ,  ;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)

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]

Agarwal, R.P.; An extension of Meijer’s G -function, Proc. Nat. Inst. Sci. India, Part A(3), (1965),536- 546. 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. Kober, H.; On fractional integrals and derivatives, Quar. J. Math., Oxford Series II, (1940),193-211. 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. Paraser, B.P.; Domain and range of fractional integration operators, Math. Japan, 12(1968), 141-145. 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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