Finite Triple Integral Representation For The Polynomial Set Tn(x 1,x 2,x 3,x 4 )

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1 IOSR Joural of Matheatics (IOSR-JM) e-issn: , p-issn: 9-765X. Volue, Issue 6 Ver. (Nov. - Dec. 05), Fiite Triple Itegral Represetatio For The olyoial Set T(x,x,x,x ) Ahad Quaisar Subhai, Brijedra Kuar Sigh, Ahad Quaisar Subhai, Dept. of Maths,Maulaa Azad College of Egg ad techology,aisabad,ata, Brijedra Kuar Sigh,Departet of Matheatics, J.. Uiversity, Chapra,Idia Abstract:-Recetly,we itroduced A uificatio of certai geeralized Geoetric polyoial Set T (x,x,x,x ),with the help of geeratig fuctio which cotais Appell fuctio of four variables i the otatio of Burchall ad Choudy [] associated with Lauricella fuctio. This geerated hypergeoetric polyoial Set covers as ay as thirty four orthogoal ad o -orthogoal polyoials.i the preset paper a attept has bee ade to express a Triple fiite itegral represetatio of the polyoial set T (x,x,x,x ). Key words:-appell fuctio, Lauricella for, Geeralized hypergeoetric polyoial. I. Itroductio The geeralized polyoial set T (x,x,x,x )is defied by eas of geeratig relatio t d e 0 ( Ar ); ( Cu ); ( Egi) F x t x t x t x t ( Bs );( Dv );( Fqi ),,, : : : : : A : C : E (.) r u g T i, : : : : : ( B ) : ( ) : ( ),,, s Dv Fq x x x x i t where (i =,,, ) ad iteger. The left had side of (.) cotais the product of two geeralized hypergeoetric fuctio which cotais Appell fuctio of four variables i the otatio of Burchaall ad Chaudy [] associated with Lauricella fuctio. The polyoial set cotais a uber of paraeters for siplicity, it is deoted by T (x,x,x,x ), where is the order of the polyoial set. After little,siplificatio(.)gives, T x x (,, x, x ) ( A ) ( B ),,,, are real ad,,,, are positive r ( ) ( ) ( ) s ( ) ( ) ( ) ( C ) ( ) ) u Eg E ) g E g D v Fq Fq Fq Eg x x x Fq!! x!!.... (.) DOI: / age

2 Fiite Triple Itegral Represetatio For The olyoial Set T (x,x,x,x ) The polyoial Set T (x,x,x,x ) happes to the geeralizatio of as ay thirty four orthogoal ad o- orthogoal polyoials. Notatios a) I. ()=... II. (A )=A.A.A..A p p III. [(A p )]=A,A,A.A p IV. [(A p )] =(A ),(A ),(A ).(A p ) V. b b b a ( ab, ),,... a a a b) I. M Theore : ( Ar) ( Cu) x ( B ) ( D )! s T x, x, x, x v, ad, we have a b c M a b c a b c s v : g : g : g : g [ :,,, ], [( a b c ) :], Fr u : q : q : q : q, [( a) :], [( b) :], [( c) :] ( ( B ) ) :,,,, ( ( D ) ) :,,,, s v ( ( A ) ) :,,,, ( ( C ) ) :,,,, r u ( rsuv) ( E ) :, g ( E ) :, g ( E ) :, ( ) : g E g ; ( F ) :, ( ) :, ( ) :, ( ) : q F q F q F q x roof :-We have ( rsuv) r r x x, x ( ) r s u v x r s DOI: / age ( rsuv) rs x d d d x (.) I a b c ( Ar) ( C ) ( ) ( ) ( ) u ( Bs ) ( Dv ) ( ) ( ) ( ) E g E g E g E g x Fq Fq F q Fq,!,

3 Fiite Triple Itegral Represetatio For The olyoial Set T (x,x,x,x ) x x a b c d d d! x!!! ( a) ( b) ( ) c a b c ( A ) r ( ) ( ) ( ) ( B ) 0 s ( ) ( ) ( ) ( Cu ) Eg E g E g E g ( Dv ) Fq Fq F q Fq x x a b c x! x!!! a b c d d d! ( Ar) ( ) ( ) ( ) ( Cu) ( B ) ( D ) s ( ) ( ) ( ) v E g E g E g E g x Fq Fq F q Fq x x a b c! x!!! a b c a b c a b c! DOI: / age

4 a b c Fiite Triple Itegral Represetatio For The olyoial Set T (x,x,x,x ) a b c ( A ) ( Bs ) ( ) ( ) ( ),,, 0 r ( ) ( ) ( ) ( ) ( ) r s u v r s u v r s x! x x! The sigle teriatig factor akes all suatio i (.) rus upto. a b c a b c T x, x, x, x O usig [6], hece the theore. xy l Where x y z articular Cases I. O takig ad y for x i (.), we achieve A x y z z dx dy dz l l (.) r 0 s u v q, g ; E, a b c y y a b c! a b c,, a b c ; F d d d a, b, c; y (ii)if we set r 0 s u v ; g q ; E( ap), F( b q) ad x i (.) we have x a b c a b c F, b; x a b c DOI: / age

5 Fiite Triple Itegral Represetatio For The olyoial Set T (x,x,x,x ), ap, a b c ; F x d d d bq, a, b, c; r 0 s v g q ; u,, C (iii) O settig ad A x y i (.) we have a b c y a b c! a b c, a b c ; F y d d d, a, b, c; r 0 s u g ; v q ; x D, F, ad x x (, ) x a b c x a b c! (iv) If we set, i (.), we get a b c,, a b c ; x F d d d, abc,, ; x (v) O takig D, F ad writig, a b c x x a b c! r 0 s u g ; v q ; ; x x for x, i (.), we get a b c (vi) O akig the substitutio, ; a b c ; x F d d d, abc,, ; x r 0 s u g ; v q ; ; x D, F ad writig x for x i (.), we get, x a b c x a b c a b c! 0 0 0,, a b c ; x F d d d, abc,, ; x DOI: / age

6 Fiite Triple Itegral Represetatio For The olyoial Set T (x,x,x,x ) r 0 s v g q ; u ; c ad x (vii) If we take for x, i (.), we get F x a b c a b c! a b c, a b c ; F x d d d, a, b, c; Refereces []. Agarwal, R.. (965):A extesio of Meijer's G-fuctio, roc. Nat. Istitute Sci. Idia,, a, []. Abdul Hali, N ad:a characterizatio of Laguerre polyoials, AI_Sala, W.A. (96) Red, Se, Uiv., padova,, []. Agarwal, B.D. ad:o a ultivariate polyoials syste []. Mukherjee, S. (988)proceedigs of atioal syposiu o special fuctios ad their applicatios. Gorakhpur (idia) p [5]. Burchall, J.L. ad:expasios of Appell's double hyper [6]. Chaudy, T.W. (9)geoetric fuctios (ii), Quart. J. Math. Oxford ser., -8 [7]. Carlitz, L. ad : Soe hyper geoetric polyoials [8]. Shrivastawa, H.M. (976)associated with the Lauricella fuctio ED of several variables [9]. Edwards, J. (96) :Treatise o itegral Calculus, vols. I, II, Chelsea ublishig Copay, New York. [0]. Erdelyi, A. (97) :Beitragzurtheorie der Kofluetehypergeoetrichefucktioe Vo ehrerebveraderliche, Akad. derwiss. Wiss. Wie. Math. Nat. v. II a, 6 (7 ad 8), p []. Rai ville,e.d. (960):Special fuctios. Mac Milla Co. New York. []. Kalla, Shyal (988) :Itegral of geeralized Jacobi; Fuctio, roceedigs of the atioal Acadey of Sci. Idia. Sac A, art I, 7 []. Szego, G. (98) :O a iequality of. Tura Cocerig Legedre polyoials, Bull. Aer. Math. Soc., 5,0-05. DOI: / age