A04 07 0105

A04 07 0105

• 1.
  International Journal ofEngineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 4, Issue 7 (December 2014) PP: 01-05 www.ijeijournal.com Page | 1 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 2 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 The H -function of two variables defined and represented by Singh and Mandia [10] in the following manner:                      1 2 2 3 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 2 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , ; , , , , ; , ,, : , : , , : , ; , , ; , , , , ; , , , , ; , j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q a A c K c e E R e Eo n m n m nx x p q p q p qy y b B d d L f F f F S H x y H H                     =   1 2 1 2 32 1 , ( ) ( ) 4 L L x y d d               (1.1) Where         1 1 1 1 1 1 1 1 1 , 1 n j j j j p q j j j j j j j n j a A a A b B                                (1.2)             2 2 2 2 2 2 1 1 2 1 1 1 1 j j n mK j j j j j j p q L j j j j j n j m c d c d                                (1.3)             3 3 3 3 3 3 1 1 3 1 1 1 1 j j n m R j j j j j j p q S j j j j j n j m e E f F e E f F                            (1.4) Where x and y are not equal to zero (real or complex), and an empty product is interpreted as unity , , ,i i i jp q n m are non-negative integers such that 0 , ( 1,2,3; 2,3)i i j jn p o m q i j      . All the 1 1 2 2( 1,2,..., ), ( 1,2,..., ), ( 1,2,..., ), ( 1,2,..., ),j j j ja j p b j q c j p d j q    3 3( 1,2,..., ), ( 1,2,..., )j je j p f j q  are complex parameters. 2 20( 1,2,..., ), 0( 1,2,..., )j jj p j q     (not all zero simultaneously), similarly 3 30( 1,2,..., ), 0( 1,2,..., )j jE j p F j q    (not all zero simultaneously). The exponents 3 2 2 3 3 3( 1,2,..., ), ( 1,..., ), ( 1,2,..., ), ( 1,..., )j j j jK j n L j m q R j n S j m q      can take on non- negative values.
• 2.
  Fractional Integration Operatorsof Certain Generalalized Hyper geometric Function...... www.ijeijournal.com Page | 2 The contour 1L is in  -plane and runs from i to i  . The poles of   2( 1,2,..., )j jd j m    lie to the right and the poles of     2 11 ( 1,2,..., ), 1 ( 1,2,..., ) jK j j j j jc j n a A j n             to the left of the contour. For 2( 1,2,..., )jK j n not an integer, the poles of gamma functions of the numerator in (1.3) are converted to the branch points. The contour 2L is in  -plane and runs from i to i  . The poles of   3( 1,2,..., )j jf F j m   lie to the right and the poles of     3 11 ( 1,2,..., ), 1 ( 1,2,..., ) jR j j j j je E j n a A j n            to the left of the contour. For 3( 1,2,..., )jR j n 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 x and y , if 1 2 1 2 1 1 1 1 0 p p q q j j j j j j j j U                 (1.5) 3 31 1 1 1 1 1 0 p qp q j j j j j j j j V A E B F             (1.6) The integral in (1.1) converges under the following set of conditions: 1 1 2 2 2 2 1 1 2 21 1 1 1 1 1 1 0 n p m q n p q j j j j j j j j j j j n j j m j j n j L K                                (1.7) 3 31 1 2 2 1 1 2 21 1 1 1 1 1 1 0 n pn p m q q j j j j j j j j j j j n j j m j j n j A A F F S E R E B                          (1.8) 1 1 | arg | , | arg | 2 2 x y     (1.9) The behavior of the H -function of two variables for small values of | |z follows as:  [ , ] 0(| | | | ),max | |,| | 0H x y x y x y    (1.10) Where 21 min Re j j m j d               21 min Re j j m j f F               (1.11) For large value of | |z ,    ' ' [ , ] 0 | | ,| | ,min | |,| | 0H x y x y x y    (1.12) Where 21 1 ' max Re j j j n j c K           , 31 1 ' max Re j j j n j e R E            (1.13) Provided that 0U  and 0V  . If we take 2 2 2 3 3 31( 1,2,..., ), 1( 1,..., ), 1( 1,2,..., ), 1( 1,..., )j j j jK j n L j m q R j n S j m q          in (2.1), the H -function of two variables reduces to H -function of two variables due to [7]. If we set 1 1 1 0n p q   , the H -function of two variables breaks up into a product of two H -function of one variable namely                2 2 3 3 1, 1, 1, 1,2 2 3 3 3 2 2 3 3 1, 1, 1, 1,2 2 2 3 3 3 : , ; , , : , ; , ,0,0: , : , 0,0: , : , : , , , ; : , , , ; j j j j j j j j j jn n p n n p j j j j j j j j j jm m q m m q c K c e E R e Em n m n x p q p q y d d L f F f F S H                
• 3.
  Fractional Integration Operatorsof Certain Generalalized Hyper geometric Function...... www.ijeijournal.com Page | 3 =                2 2 3 3 1, 1,1, 1, 3 3 32 2 2 2 2 3 3 1, 1, 1, 1,2 2 2 3 3 3 , ; , ,, ; , ,, , , , , , , ; , , , ; j j j j jj j j j j n n pn n p j j j j j j j j j jm m q m m q e E R e Ec K cm n m n p q p q d d L f F f F S H x H y                  (1.14) If 0  , we then obtain                    1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , ; , , , , ; , ,0, : , : ,2 , : , : , , ; , , , , ; , , , , ; j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q a A c K c e E R e En m n m n x p q p q p q y b B d d L f F f F S H                        =                    1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , ; , , , , ; , ,0, : , : , , : , : , , ; , , , , ; , , , , ; j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q a A c K c e E R e En m n m n x p q p q p q y b B d d L f F f F S H                 (1.15)                    1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , ; , , , , ; , ,10, : , : , , : , : , 1 , ; , , , , ; , , , , ; j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q a A c K c e E R e En m n m n x p q p q p q b B d d L f F f F S y H                 =                    1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 , ; , 1 , , 1 , ; , 1 , , 1 , ;0, : , : , , : , : , 1 , ; , 1 , ; , 1 , , 1 , ; , 1 , j j j j j j j j j j j j jq m m q m m q j j j j j j j j j j j j jp n n p n n p b B d d L f F f F Sn m n m n x p q p q p q y a A c K c e E R e E H                           (1.16) II. Definitions We introduce the fractional integration operators by means of the following integral equations:  , ; 1 , ; 0 ( ) ( ) x r r r r U x UY f x rx t x t f t H dt                     (2.1) And  , ; 1 , ; 0 ( ) ( ) x r r r r V x VN f x rx t t x f t H dt                     (2.2) Where U and V represent the expressions 1 m nr r r r t t x x             and 1 m nr r 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 1 1 , , 1p q p q      (ii) 1 Re , j j j j d f rm rm F q             1 Re , j j j j d f rn rn F q             1 Re , j j j j d f rm rm F p             (iii) ( ) (0, ).pf x L  The last condition ensures that Y and N both exists and also that both belongs to (0, )pL  . If we set 1 1 1 0n p q   , we obtain the operators involving the product of two H -functions. If we take 2 2 2 3 3 31( 1,2,..., ), 1( 1,..., ), 1( 1,2,..., ), 1( 1,..., )j j j jK j n L j m q R j n S j m q          we obtain the operators involving H -function of two variables.
• 4.
  Fractional Integration Operatorsof Certain Generalalized Hyper geometric Function...... www.ijeijournal.com Page | 4 III. Mellin Transform The Mellin transform of ( )f t will be defined by [ ( )]M f t . We write 1 s p it   where p and t are real. If 1, ( ) (0, )pp f t L   , then 1 0 1, [ ( )] ( )s p M f t t f t dt      (3.1) 1 1/ 1, [ ( )] . . . ( ) x s x p M f t l i m t f t dt    (3.2) Where . . .l i m denotes the usual limit in the mean for pL -spaces. Theorem 1. If 1 1 ( ) (0, ),1 2[ ( ) (0, ) 2]| arg | ,| arg | ,( , ) 0 2 2 p pf x L p or f x M and p                  1 Re , j j j j d f rm rm F q             1 Re , j j j j d f rn rn F q             then   1 2 2 3 2 1 1 2 2 2 2 , : , : ,, ; 2, 2: , ; ,, ; ( ) o n m n m nr p q p q p qxM Y f x H                            1, 1, 1, 1, 1,1 2 2 2 3 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , , , ; , , , , ; , , 1 , ; , , , , , , ; , , , , ; [ ( )] j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q s a A m c K c e E R e E x r y s b B m n d d L f F f F S r M f x                                     (3.3) Where (0, )pM  denotes the class of all functions ( )f x of (0, )pL  with 2p  which are inverse Mellin transforms of functions of ( , )qL   . Proof: From (2.1), it follows that  , ; , ; ( )r xM Y f x        1 1 0 0 ( ) x s r r r U Ux rx t x t f t H dtdx                =  2 0 0 ( ) ( ) x s r r r U Ut f t dt x x t f t H 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                    1 2 2 3 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 2 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; , , ; , , , , ; , ,, : , : ,1 1 (1 ) , : , ; , (1 ) , ; , , , , ; , , , , ; (1 ) k l j j j j j j j j j j j j jp n n p n n p k l j j j j j j j j j j j j jq m m q m m q a A c K c e E R e Eo n m n m n x x p q p q p q x x b B d d L f F f F S x x H dx                          1 0  =                    1 2 2 3 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 1 2 2 2 2 1, 1, 1, 1, 1,1 2 2 2 3 3 3 , ; ,(1 , ), , ; , , , , ; , ,, : , : , 2, 2: , ; , , ; ,(1 , ), , , , ; , , , , ; j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q a A k c K c e E R e Eo n m n m n p q p q p q b B k l d d L f F f F S H                            (3.4) Where Re 0,Re 0 j j j j j j j j d f d f k k k k F F                         1 1 | arg | ,| arg | ,( , ) 0 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 ( ) (0, ),1 2[ ( ) (0, ) 2]| arg | ,| arg | ,( , ) 0 2 2 p pf x L p or f x M and p                  1 Re , j j j j d f rn rn F q             1 1 1 Re , 1 j j j j d f r rm rm F q p q                 then
• 5.
  Fractional Integration Operatorsof Certain Generalalized Hyper geometric Function...... www.ijeijournal.com Page | 5   1 2 2 3 2 1 1 2 2 2 2 , : , : ,, ; 2, 2: , ; ,, ; ( ) o n m n m nr p q p q p qxM N f x H                            1, 1, 1, 1, 1,1 2 2 2 3 3 3 1, 1, 1, 1, 1,1 2 2 2 3 3 3 1 , ; , , , , ; , , , , ; , , , ; , , , , , , ; , , , , ; [ ( )] j j j j j j j j j j j j jp n n p n n p j j j j j j j j j j j j jq m m q m m q s a A m c K c e E R e E r s b B m n d d L f F f F S r M f x                                      (3.5) Theorem 3. If 1 1 ( ) (0, ), ( ) (0, ),| arg | ,| arg | ,( , ) 0 2 2 p pf x L g x L               , 1 1 Re max , , j j j j d f rm rm F p q                 1 1 Re 0, 1 j j j j d f r rm rm F p q              then , ; , ; , ; , ; 0 0 ( ) ( ) ( ) ( )r r x xg x Y f x dx f x N 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). 1 1 , ; 1 , ; , ;, ; ( ) ( )r r xx x Y f x N f x                (4.1) 1 1 , ; 1 , ; , ;, ; ( ) ( )r r xx x N f x Y f x                (4.2) , ; , ; , ; , ;( ) ( )r r x xx Y f x Y x f x                   (4.3) , ; , ; , ; , ;( ) ( )r r x xx N f x N x f x                   (4.4) If , ; , ; ( ) ( )r xY f x g x       , then , ; , ; ( ) ( )r xY f cx g cx       (4.5) And if , ; , ; ( ) ( )r xN f x x        , then , ; , ; ( ) ( )r xN 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] Agarwal, R.P.; An extension of Meijer’s G -function, Proc. Nat. Inst. Sci. India, Part A(3), (1965),536- 546. [2] Bromwich, T.J.T.A.; Theory of Infinite Series, Macmillan, London (1954). [3] Erdelyi, A.; On fractional integration and application to the theory of Hankel transforms, Quar. J. Math., Oxford Series 2(44), (1940),293-303. [4] Kober, H.; On fractional integrals and derivatives, Quar. J. Math., Oxford Series II, (1940),193-211. [5] Kalla, S.L. and Saxena, R.K.; Integral operators involving hypergeometric functions, Math. Zeitscher, 108, (1969), 231-234. [6] Lowndes, J.S.; A generalization of the Erdelyi-Kober operators, Proc. Edinburgh Math. Soc. 17(2), (1970), 139-148. [7] Mittal, P.K. & Gupta, K.C.; An integral involving generalized function of two variables. Proc. Indian Acad. Sci. Sect. A( 75)1961), 67-73. [8] 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. [9] Paraser, B.P.; Domain and range of fractional integration operators, Math. Japan, 12(1968), 141-145. [10] 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 [11] Saxena, R.K.; On fractional integration operators, Math. Zeitcher, 96(1967), 288-291. [12] 12. Saxena, R.K.; Integrals of products of H -functions, Universidad Nac. De. Tuc. Rev. Ser. A(1971),185-191. [13] 13. Saxena, R.K. and Kumbhat, R.K.; A generalization of Kober operators, Vijnana Parishad Anusandhan Patrika, India, 16(1973), 31-36. [14] 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.