Generating Functions for Laguerre Type Polynomials. Group Theoretic method

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1 It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet of Mathematics, S.V. Natioal Istitute of Techology, Surat-95007, Idia ** PAE, Aad Agricultural Uiversity, Muvaliya Farm, Dahod-89151, Gujarat, Idia ajayshula2@rediffmail.com shriat_math@rediffmail.com Abstract A attempt is made to obtai geeratig relatios of modified Laguerre polyomials. i two-variables by meas of Group theoretic method. Laguerre polyomials have special importace i egieerig, scieces ad costitute good model for may systems i various fields. Mathematics Subject Classificatio: C45; C50; C80 Keywords: Two-variable Laguerre polyomials, Recurrece relatios, Group Theoretic Method ad geeratig relatio 1. INTRODUCTION AND PRELIMINARIES: The Group theoretical aalysis provides us a effective tool for fidig geeratig fuctio of special fuctios. The above idea was origially geerated by Weiser [9]

2 258 A. K. Shula ad S. K. Meher ad he [10 ad 11] also applied this techique to obtai the geeratig relatio. Miller, McBride, Srivastava ad Maocha [4, ad 6] respectively reported Group theoretic method for obtaiig geeratig relatios i their boos. S. Kha, M.A. Patha ad G. Yasmi [2] studied represetatio of a Lie algebra G(0,1) ad three variable geeralised Hermite polyomials, H ( xy,, ). S. Kha ad G. Yasmi [2] have also doe some wors o Laguerre polyomial of two variables by usig Group theoretic method. S. Kha ad M.A. Patha [] studied a paper o Lie-theoretic geeratig relatios of Hermite 2D polyomials. Agai they studied Lie theory ad two variable geeralised Hermite polyomials Group theoretic method basically gives a coectio betwee special fuctio ad the matrix groups. I preset paper our aim is to study first order liear differetial operator which geerates Lie algebra isomorphic to some matrix Lie algebra ad apply these operators to determie a local represetatio [ T( g) f ] which maes a oe to oe correspodece betwee these two Lie algebras., Afterwards by choosig the suitable values of f(x, y) the above represetatio leads us to geeratig fuctios. Cosider the abstract group G ( 0,1) cosists of all 4 4 matrices of the form t 1 ce a t t 0 e b 0 g = (1.1) where the group operatio is matrix multiplicatio. Now we ca itroduced coordiates for the elemets g i G ( 0,1) by settig g abct,,,. (1.2) ( ) Thus G ( 0,1) is a complex 4-dimesioal Lie group. Here the coordiates (1.2) are valid over the etire group ad the group G ( 0,1) is simply coected. The Correspodig Lie algebra of the Lie group G ( 0,1) is LG( 0,1) all 4 4 matrices of the form α 0 x2 x4 x 0 x x αβ, L G 0,1. 1 = with the Lie product [ α, β] αβ βα = for ( ) This is of dimesio 4 with the basis B = { ε} = G4 the space of, x1, x2, x, x4 (1.),,,.

3 Geeratig fuctios for Laguerre type polyomials 259 where the taget matrices correspodig to basis have respectively the forms = , =, = ad ε = They satisfy the commutatio relatios + +, =,, =,, = ε ε = ε = ε =,,, 0 (1.4) (1.5) where 0 is the 4 4 ero matrix, form a basis for LG( 0,1) The expoetial map expα, α L[ G( 0,1) ], is of the form expα = e α = = α 0!. ad it is a aalytic diffeomorphism mappig all of L G( 0,1) oto ( 0,1) I particular a 0 exp( a + ) =, c c 0 e 0 0 exp( c ) = ad G. 1 b exp( b ) =, d exp( dε ) = α L x, y is defied as: Let the Laguerre type polyomial of two variables ( ) r r α y Γ ( α + 1)( x) 1 t L ( x, y) =, where ( ) t e dt ( ) r= 0 ( r)! r! Γ ( α + r + 1) ad its geeratig fuctio is give by as 0 (1.6) Γ =,Re > 0 (1.7)

4 260 A. K. Shula ad S. K. Meher 1 α L ( x, y) t = exp α + 1 = 0 ( 1 yt) 1 yt (1.8) where α is a o egative iteger. α L xy, satisfy the differetial equatio The polyomial ( ) 2 d x d α x + 1 L 2 α + ( x, y) = 0 (1.9) dx y dx y xt 2. GROUP THEORETIC METHOD: Now replacig d dx by x differetial equatio ad by i equatio (1.9) we costruct a partial 2 2 x x + 1 f 2 α + ( x, y, ) = 0 (2.1) x x y x y Thus by [observatio 1, pp.27] of H. M. Srivastava ad H. L. Maocha [6], α (,, ) α f xy = L ( xy, ) is a solutio of equatio (.1) sice L ( xy, ) is a solutio of equatio (2.1). + Cosider the first order liear differetial operators J, J, J ad E defied by J = + 2 x J = x + α (2.2) x y y J = x ad E = 1 These operators satisfy the commutatio relatios + + J, J = J, J, J = J ad J, J = E. (2.) EJ, = EJ, = EJ, = 0 This is idetical with the commutatio relatio (1.5) for the geerators of G (0,1). Thus by + Theorem 1.10 of Miller W. Jr. [4], the operators J, J, J ad E geerates the Lie algebra of geeralied Lie derivatives. Agai these operators also satisfy the followig properties + α α + 1 J L ( x, y) = ( + 1 ) L+ 1 ( x, y) α α 1 J L ( x, y) = αl 1 ( x, y) (2.4)

5 Geeratig fuctios for Laguerre type polyomials 261 α (, ) (, ) α (, ) (, ) α J L x y = L x y α E L xy = L x y I terms of the J operators, we itroduced the Casimir operator [p.2] of Miller W. Jr. [4], C = J J EJ 2 2 y x 1 x = + α 2 + x x y x y Hece, 2 2 C y x 1 x = + α 2 + (2.5) x x y x y We ca easily verified from the commutatio relatio (2.), that C commutes with + J, J, J ad E. CJ, = CJ, = CJ, = CE, = 0 (2.6) by usig equatio (2.5),we ca write equatio (2.1) as 2 2 x C f ( x, y; ) = y x + α 1 f 2 + ( x, y; ) x x y x y = 0 (2.7) Now o proceedig to compute the multiplier represetatio g f x, y, g G 0,1 iduced by the J- operators. [ ] [ T ( ) ]( ), ( ) I order to determie the multiplier represetatio T of G ( 0,1), we first compute the + actio of exp( aj ) f, exp( bj ) + To obtai exp( aj ) f f, ( ) exp cj f ad exp( d E) f. by usig [Theorem -7] of H. M. Srivastava ad H. L. Maocha [6], we eed followig facts d x ( a ) = x ( a ) ( a ) da (2.8) d y ( a ) = 0 da (2.9) d ( a) = { ( a) } 2 da (2.10) d x( a) v( a) = v( a) α ( a) da y( a) (2.11) x 0 x y 0 y 0 v 0 = 1. ad satisfyig the coditio ( ) =, ( ) =, ( ) = ad ( ) as, exp ( aj + ) f ( x, y; ) = v( t) f ( x( a), y( a); ( a) ),

6 262 A. K. Shula ad S. K. Meher where x( a ), ya ( ) ad a ( ) are the solutios of the aforesaid equatio. O solvig the above differetial equatio ad by usig iitial coditios, we get T( exp ( a + α )) f ( x, y; ) ( 1 ) exp ax = a f x( 1 a), y; (2.12) y 1 a where a < 1. Similarly T( exp ( b )) f by ( x, y; ) = f x, y; + (2.1) T( exp ( c )) f ( x, y ; ) = f ( xy, ; exp( c )) (2.14) ad T( exp ( d ε )) f ( x, y; ) = exp ( d) f ( x, y; ) (2.15) These are all defied for a, b, c ad d sufficietly small. Now, ( exp + ( ) exp T a ( b ) exp ( c ) exp ( dε )) f ( x, y ; ) = T( exp( a )) T( exp( b )) T( exp( c )) T( exp ( dε )) f ( x, y; ) α ax by = ( 1 a) exp + d f x + ( 1 a), y ; exp( c) y Let g = exp( a )exp( b )exp( c ) exp ( dε ), g G4 By usig (1.6) we ca write i.e c 1 be d c c 0 e a 0 exp( a )exp( b )exp( c ) exp( dε ) = (2.16) α ax by [ T ( g) f ]( x, y; ) = ( 1 a) exp + d f x + ( 1 a), y ; exp( c) y (2.17) Where c 1 be d c c 0 e a 0 g = G (2.18) f x, y ; = α L x, y which is a commo Eige fuctio of C ad J. Settig ( ) ( )

7 Geeratig fuctios for Laguerre type polyomials 26. GENERATING RELATIONS: α To accomplish our tas for obtaiig geeratig fuctios of L ( xy, ) fuctio f ( x, y; ) which satisfies the differetial equatio (1.9). Cosider f ( x, y; ) is a commo eigefuctio of C ad, we fid a J, ad also a solutio of the simultaeous equatio Cf( xy, ; ) = 0 (.1) J f ( x, y; ) = f ( x, y; ) (.2) Equatio (.1) ad (.2) ca also be writte as 2 2 x y x + 1 f 2 α + ( x, y; ) = 0 (.) x x y x y f ( x, y; ) = 0 (.4) respectively. By usig (.) ad (.4), we arrive at coclusio that f ( x, y; ) = L α ( x, y) O applyig the (observatio, p.24, [8]), we have α ax α [ ( ) ]( ) ( ) by T g f x, y; = 1 a exp + d L ( ) ( ( )) x + 1 a, y exp c y α ax α by = ( 1 a) exp + c + d L x + ( 1 a), y, a p 1 y (.5) This satisfies the relatio CT( g) f ( x, y; ) = 0 (.6) If ad α are o egative iteger the (.6) ca be writte i the followig form If [ T ( g) f ]( x y; ) [ T ( g) f ]( x y; ) = P ( g, x, y ) +, (.7) =, is regular at x = 0 ad by usig (observatio 2, p.24, [8]) the we get, α P g, x, y = A g L x y (.8) ( ) ( ) ( ) +, α Therefore, [ T ( g) f ]( x, y; ) = A ( g) L ( x, y ) From (.5) ad (.9) we get, = = = 0 A + α ( g) L ( x, y ) + (.9)

8 264 A. K. Shula ad S. K. Meher ax y α α ( 1 a) exp + c + d L x + ( 1 a), y = by = α ( g) L ( x y ) A, + + (.10) O solvig the above equatio we get, Γ ( ) ( ) ( 1+ + ) 1 α + + 1, ; A g = 1 a { Γ( 1+ ) } 2F1 ( ) ab Γ 1+ (.11) 1+ ; By substitutig the value of A ( g) i equatio (.10) we get the required geeratig fuctio as: ax α α by Γ ( ) ( ) ( + + ) a L x + a y = 1 α 1 exp 1 1, L+ ( x, y) { Γ( 1+ ) } y = Γ( 1+ ) α + + 1, ; F ab ( a) ; where 1 0 p p a. (.12) SPECIAL CASES: For b = 0 ad = 1, it becomes ax α ( ) ( ( ) ) ( ) a L ( )( ) x a y α 1 α exp 1+ 1+, = L+ x, y a y = 0! while for a = 0 ad = 1, we have (( ) ) α b α L x by, y = L+ ( x, y) = 0! if is a positive iteger say = m, the geeratig fuctio followig from (.5) is ax α α m by! α 1 exp ( 1 a) Lm x + ( 1 a), y = L ( x, y) { Γ( 1+ m) } y = 0 m! α + + 1, ; 2F1 ab a m 1+ ; 5. CONCLUSION: m ( ) (.1) (.14) (.15) I this wor, the Group theoretic method has bee successfully applied to Modified α Laguerre polyomial i two variables L ( xy, ) ad this method is easy ad straight forward for obtaiig the geeratig relatios. The reaso of iterest for this family of Laguerre polyomial is due to their itrisic mathematical importace ad to the fact that

9 Geeratig fuctios for Laguerre type polyomials 265 these polyomials are show to be atural solutios of a particular set of partial differetial equatios which ofte appear i the treatmet of radiatio physics problems such as the electromagetic wave propagatio ad quatum beam life-time i storage rig. REFERENCES [1] G. Dattoli, Geeralised Polyomials, Operatioal Idetities ad Their Applicatios, Joural of Computatioal ad Applied Mathematics, 118 (2000), [2] S. Kha, M.A. Patha, G.Yasmi, Represetatio of a Lie algebra G (0, 1) ad Three Variable Geeralised Hermite Polyomials, H (x, y, ), Itegral Trasforms Special Fuctios, 1 (2002), [] S. Kha, M. A. Patha, Lie-Theoretic Geeratig Relatios of Hermite 2D Polyomials, Joural of Computatioal ad Applied Mathematics, 160 (200), [4] S. Kha, G. Yasmi, Lie-Theoretic Geeratig Relatios of Two Variable Laguerre Polyomials, Reports o Mathematical Physics, 51 (200), 1-7. [5] E. B. McBride, Obtaiig Geeratig Fuctios, Spriger Verlag, Berli, [6] W. Jr. Miller, Lie Theory ad Special Fuctios: Academic Press, New Yor ad Lodo, [7] N. K. Raa, Lie Theoretic Origi of Some Geeratig Fuctios for the Laguerre Polyomials, Bull. Cal. Math. Soc., 94 (2002), [8] H. M. Srivastava ad H. L. Maocha, A Treatise o Geeratig Fuctios, Ellis Horwood Ltd.Chichester, [9] L. Weiser, Group-theoretic Origi of Certai geeratig fuctios, Pacific J. Math. 5 (1955), [10] L. Weiser, Geeratig fuctios for Hermite fuctios, Caad. J. Maths, 11 (1959), [11] L. Weiser, Geeratig fuctios for Bessel fuctios, Caad. J. Maths, 11 (1959),

10 266 A. K. Shula ad S. K. Meher Received: May, 2010