A Generalization of Hermite Polynomials

Transcription

1 Ieraioal Mahemaical Forum, Vol. 8, 213, o. 15, HIKARI Ld, A Geeralizaio of Hermie Polyomials G. M. Habibullah Naioal College of Busiess Admiisraio & Ecoomics Gulberg-III, Lahore, Pakisa musafa1941@yahoo.com Abdul hakoor Naioal College of Busiess Admiisraio & Ecoomics Gulberg-III, Lahore, Pakisa chshakoor@gmail.com Coyrigh 213 G. M. Habibullah ad Abdul hakoor. This is a oe access aricle disribued uder he Creaive Commos Aribuio Licese, which ermis uresriced use, disribuio, ad reroducio i ay medium, rovided he origial work is roerly cied. Absrac The ieded objecive of his aer is o exed he Hermie olyomials based o hyergeomeric fucios ad o rove basic roeries of he exeded Hermie olyomials. Mahemaics ubjec Classificaio: 33C45, 5A15, 11B37. Keywords: Orhogoal olyomials, Hermie olyomials, Hyergeomeric fucios, Geeralized hyergeomeric fucios. 1. Iroducio Hermie Polyomials ad is alicaios have bee sudied for log ad sill arac aeio. Oe ca refer a log lis of books ad jourals for advaced kowledge of Hermie olyomials ad is exesios, for examle [7] ad [6], for books ad [2], [4], [5], [7], [8], [9], [1] ad [11] for jourals. Based o a geeralized hyergeomeric fucio, we iroduce here a geeralizaio of he Hermie olyomials ha rovide aural exesios of basic resuls ivolvig he Hermie olyomials as a sudy of he Laguerre olyomials i [3].

2 72 G. M. Habibullah ad Abdul hakoor x geeraed by he fucio Gx (, ) = ex( x ) is o be kow as he geeralized Hermie olyomial se. Noe ha for = 2, i reduces o he kow geeraig fucio for he Hermie olyomials. We firs deduce a exlici exressio for his geeralized Hermie olyomials. For a osiive ieger, he se {, ( )} Theorem 1: For a o-egaive ieger ad a osiive ieger, we have, Proof: By cosiderig oe ha e k k (1.1) k =!! ( 1 )! k ( k) = =,! = e ( x ) ( x ) ( ) k = =! k= k! k + k ( 1) = k!! = k= A use of a variaio of Lemma of [6] wih i lace of 2 leads o k k, ( ) x 1 = = k= =,! k!( k)!, which imlies ha, k k k = ( 1 )! k ( k) =!!

3 A geeralizaio of Hermie olyomials 73 We ow deermie a few recurrece relaios for he geeralized Hermie olyomials. Theorem 2: For all fiie x,, a osiive ieger ad a o-egaive ieger, x = 1 2 L + 2, (1.2) (i) ( )( ) ( ),,, + 1 ( + ), ( ) ( )( ) L( ) (ii) 2 Proof: = x Cosider ( ),, + 2 F = G x = f. = Takig arial derivaives of F w.r. x ad, we have F F 1 = G, = ( x ) G. x Also, + 1 xf x f x f x = = = ( ) ( ) ( ) =. (1.3) These relaios give rise o 1 F F ( x ) =, (1.4) x ad cosequely xf f = f 1. + ice by akigg = e x,, f( x ) =, we fially ge! ( )( ) ( ) x = 1 2 L + 2.,,, + 1 imilarly, we ca rove (ii). Theorem 3: For ay real umber c ad a osiive ieger, we have

4 74 G. M. Habibullah ad Abdul hakoor = ( ) c, c () i = ( 1 x)! c c+ 1 c+ 2 c+ 1 F,,, K, ; ; 1, x ( ) ( ) ( ii), = F,,,, ; ;, K x (1.6) ( iii) x k =, k! =. (1.7) k! k! Proof: Noe ha so ha ( ), ( ) c x ( 1 ) ( c) =! k! ( k)! ( 1) ( ) ( ) = = k= = = k= k k k + k c x + k k!! k k = k= =! k!, k ( c) ( 1, ) ( c) =! k k!1 ( x) = = k ( c+ k) ( 1) ( c) k k c+ k, which imlies ha = ( ) c,! ( 1 x) c = c c+ 1 c+ 2 c+ 1 F,,, K, ; ; 1 1 x

5 A geeralizaio of Hermie olyomials 75 arig wih =, k k k = k! 1 ( ) k! k!, ad e x, = e, we ca rove (ii) ad (iii). =! Followig radiioal heory, we ca rove orhogoaliy, iegrals ad exasios ivolvig he Hermie olyomials ad is relaios wih oher olyomials. We ca also cosider q Hermie olyomials ad rove corresodig resuls. Refereces [1] A. Ali ad E. Erkus, O a mulivariable exesio of he Lagrage- Hermie olyomials, Iegral Trasforms ad ecial Fucios, (26), [2] A. J. Dura, Rodrigue's formulas for orhogoal marix olyomials saisfyig higher-order differeial equaios. Exerimeal Mahemaics, 2 (211), [3] A. Kha ad G. M. Habibullah, Exeded Laguerre olyomials. I. J. Coem. Mah. ci., 22 (212), [4] C. Berg ad A. Ruffig, Geeralized q -Hermie olyomials. Comm. Mah. Phys., 223 (21), [5] C. Kaaoglu ad M. A. Ozarsla, ome roeries of geeralized mulile Hermie olyomials. J. Com. Al. Mah., 235(211), [6] E. D. Raiville, ecial Fucios, The Macmilla Comay. New York, 196. [7] G. Adrews, R. Askey ad R. Roy. ecial Fucios, Cambridge Uiversiy Press, [8] H. Chaggara ad W. Koef, O liearizaio ad coecio coefficies for geeralized Hermie olyomials, J. Com. Al. Mah., 236 (211),

6 76 G. M. Habibullah ad Abdul hakoor [9] K. Y. Che ad H. M. rivasava, A limi relaioshi bewee Laguerre ad Hermie olyomials. Iegral Trasforms ad ecial Fucios, 16 (25), [1] R.. Baaha, A ew exesio of Hermie marix olyomials ad is alicaios. Liear Algebra ad is Alicaios, 419 (26), [11]. B. Trickovic ad M.. akovic, A ew aroach o he orhogoaliy of he Laguerre ad Hermie olyomials. Iegral Trasforms ad ecial Fucios, 17 (26), Received: Jauary 11, 213