International Journal of Engineering 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 Singh 1 , Nanda Kulkarni 2 (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 j p n n p n n p j j j j j j j j j j j j j q m m q m m q a A c K c e E R e E on m n m n x x p q p q p q y y b B d d L f F f F S Hxy H H                     =   1 2 1 2 3 2 1 , () () 4 LL xydd          (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 m K 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 j pqnm are non-negative integers such that 0 , ( 1, 2, 3; 2, 3) i i j j n po m q i j       . All the 1 1 2 2 ( 1, 2,..., ), ( 1, 2,..., ), ( 1, 2,..., ), ( 1, 2,..., ), j j j j a j p b j q c j p d j q     3 3 ( 1, 2,..., ), ( 1, 2,..., ) j j e j p f j q   are complex parameters. 2 2 0( 1, 2,..., ), 0( 1, 2,..., ) j j j p j q       (not all zero simultaneously), similarly 3 3 0( 1, 2,..., ), 0( 1, 2,..., ) j j E 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 j K j n L j m q R j n S j m q       can take on non- negative values.