Shivley s Polynomials of Two Variables
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1 It. Joural of Math. Aalysis, Vol. 6, 01, o. 36, Shivley s Polyomials of Two Variables R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Departmet of Mathematics Sardar Vallabhbhai Natioal Istitute of Techology Surat, Gujarat, Idia rkjaa003@yahoo.com ibrahimmaths@gmail.com ajayshukla@rediffmail.com Abstract I this preset paper we itroduced Shivley s Polyomials of two variables i the same lie as Pseudo Laguerre Polyomials of oe variable as defied by Shivley (see 5 pp. 98. The Geeratig Fuctios ad Operatioal Represetatios of these ew polyomials are also obtaied. Some special cases of the established results are also deduced as corollaries. Mathematics Classificatio: 33C45, 33C99, 34A5 Keywords: Pseudo Laguerre Polyomial, Geeratig Fuctio, Operatioal Represetatio 1. INTRODUCTION The classical Laguerre polyomial L (α Hypergeometric fuctio as (x is defied by 1, 5 i the form of L (α (x (α +1 1F 1 ( ; α +1;x (1! It is ecessary to distiguish betwee the case i which α is idepedet of, we shall call L (α (x a proper Laguerre polyomial, ad whe α depeds o, we shall call L (α (x a Pseudo Laguerre Polyomial. Shivley (see 5 pp. 98 has defied the polyomial R (a (x by the equatio R (a (x (a + 1F 1 ( ; a + ; x (!
2 1758 R. K. Jaa, I. A. Salehbhai ad A. K. Shukla i which is ay o-egative iteger, ad a is idepedet of. (x may also be writte as R (a R (a (x (a (a! 1 F 1 ( ; a + ; x (3 Some facts are elisted below for our ivestigatio: Geeratig Fuctio 4: Let G (x, t be a fuctio that ca be expaded i powers of t such that G (x, t c f (x t (4 0 where c is a fuctio of that may cotai the parameters of the set {f (x}, but is idepedet of x ad t. The G (x, t is called a Geeratig Fuctio of the set {f (x} as defied i 4. Series Maipulatio Techiques 5: I the study of geeratig fuctios of the sets of polyomials, we sometimes deal with diverget power series. The maipulative techiques are useful whe applied to coverget series. The followig two basic lemmas give i 5 are useful. Lemma 1. A(k, A(k, k (5 ad Lemma. 0 k0 0 k0 B(k, B(k, + k (6 0 k0 0 k0 A(k, A(k, k (7 0 k0 0 k0 ad B(k, B(k, +k (8 0 k0 0 k0 Kummer s Secod Formula 5: If a is ot a odd iteger < 0, the ( e z 1F 1 (a;a;z 0 F 1 ; a + 1 ; z (9 4
3 Shivley s polyomials of two variables 1759 Operatioal Represetatio : Kha ad Shukla 3, recetly Kha et al. applied operatioal techiques to obtai certai polyomials as: If D x ad D x y, oe ca write the biomial expasio for (D y x + D y as (D x + D y where, C r! ( r!. O writig the series i reverse order oe ca write (10 as (D x + D y C r D r x D r y, (10 C r D r xd r y (11 Now, if f(x is a fuctio of x aloe ad g(y is a fuctio of y aloe, the from (10 ad (11 we have, (D x + D y {f(xg(y} (D x + D y {f(xg(y} ( r ( 1 r Dx r {f(x} Dy r {g(y} (1 ( r ( 1 r Dx r {f(x} D r y {g(y} (13 Similarly, if D x x y y ad D z ; the triomial expasio for z (D x + D y + D z is give by ad (D x + D y + D z (D x D y + D y D z + D z D x r s0 From (14 ad (15, it is clear that (D x + D y + D z {f(xg(yh(z} r s0 ad ( r+s ( 1 r+s s! r s0 ( r+s ( 1 r+s s! ( r+s ( 1 r+s s! Dx r s {f(x} Dy r {g(y} Dz s {h(z} (D x D y + D y D z + D z D x {f(xg(yh(z} D r s x D r y Ds z (14 D s x Dy r D r+s z (15
4 1760 R. K. Jaa, I. A. Salehbhai ad A. K. Shukla r s0 ( r+s ( 1 r+s s! Dx s {f(x} Dy r {g(y} Dz r+s {h(z}. PSEUDO LAGUERRE POLYNOMIALS OF TWO VARIABLES We itroduce Shively s pseudo Laguerre polyomials of two variables as: R (α (x, y (α +! r s0 ( r+s x s y r (α + s s! (16 3. GENERATING FUNCTION FOR R (α (x, y: Shivley deduced the Geeratig Fuctio for Pseudo Laguerre polyomials of oe variable as 0 R (α (x t ( e t 0F 1 ; α +1 ; t xt (17 Proceedig i the same way, we had deduced the Geeratig Fuctio for Pseudo Laguerre polyomials of two variables. THEOREM 3.1: The geeratig fuctio of R (α (x, y is give by 0 R (α (x, y t where R (α (x, yis defied by (16. Proof: Now, R (α t (x, y 0 0 ( α 0 s0 (α! e t Further simplificatio yields 0 R (α (x, y e t 0F 1 ; α +1 ; t (1 y xt ( α+1 0 t (α+! t r s0 r s0 ( r+s x r y s (α + r s! (4t +r+s ( ( α α + r r +s ( 1 r+s x r y s (α r (α +r +s!s! 0 {t (1 y} t! ( xt r +r e t {t (1 y xt} 0! ( α+1 e t 0F 1 ; α +1 ; t (1 y xt ( r+s x r y s (α+ r s! (18
5 Shivley s polyomials of two variables 1761 Which is the required geeratig fuctio for the fuctio R (α (x, y. Corollary 3.1: It may be of iterest to poit out that the above geeratig fuctio implies that R (α (x, 0 R(α (x. Proof: By puttig y0 i (18, we will get, R (α (x, 0 t ( α+1 e t 0F 1 ; α +1 ; t xt 0 Which is exactly the geeratig fuctio of R (α (x as depicted i (17 i.e. R (α (x, 0 t ( α+1 e t 0F 1 ; α +1 ; t xt R (α 0 (x t ( α+1 0 which implies that R (α (x, 0 R (α (x Hece the proof. Corollary 3.: Agai, from (18 we ca deduce that R (α (0,y 1! F ; +1 ; y 1 1 α ;. (19 Proof: By puttig x0 i (18, we will get, 0 R (α (0,y t e t 0F 1 ; α +1 ; t (1 y e t (t (1 y 0! ( α+1 e t (t 0! ( ( k y k α+1 k0 k! Further simplificatio yields R (α (0,y t ( α+1 0 From (0, it is clear that, R (α (0,y 1! F 1 Hece the proof. 1! F 1 t ( α+1 ; +1 0 ; y 1 α ; ; y 1 α ; 4. OPERATIONAL REPRESENTATION OF R (α (x, y: (0
6 176 R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Kha ad Shukla 3 ad recetly Kha et al. applied the operatioal represetatio of certai polyomials of several variables, proceedig i the same way as metioed i 4,6 ad usig (1.9, we ca also obtai the operatioal represetatio for R (α (x, yas: (D x D y + D y D z + D z D x {x α+ 1 y e z } r s0 r s0 ( r+s ( 1 r+s s! ( r+s ( 1 r+s s! { (! x α+ 1 e z (α+! (! x α+ 1 e z R α Dx s {x α+ 1 } Dy r {y } Dz r+s {e z } (x, y (α + 1!! xα+ 1 s (α + 1 s! yr ( 1 r+s e z r s0 } ( r+s x s y r (α+ s s! (D x D y + D y D z + D z D x { x α+ 1 y e z} (! x α+ 1 e z R α (x, y (1 Which is the required operatioal represetatio for R (α (x, y. REFERENCES 1. G. E. Adrews, R. Askey ad R. Roy, Special Fuctios, Cambridge Uiv. Press, Cambridge, M. A. Kha, A. H. Kha ad N. Ahmad, A ote o Hermite Polyomials of Several Variables, Applied Mathematics ad Computatio, Vol. 18, No. 11 (01, M. A. Kha ad A. K. Shukla, O Biomial ad Triomial Operator Represetatio of Certai Polyomials, Italia Joural of Pure ad Applied Mathematics, Vol. N-5 (009, E. B. McBride, Obtaiig Geeratig Fuctio, Spriger-Verlag, New York, E. D. Raiville, Special Fuctios, The Macmilla Compay, New York, Received: March, 01
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