Extended Laguerre Polynomials

Transcription

1 I J Coemp Mah Scieces, Vol 7, 1, o, Exeded Laguerre Polyomials Ada Kha Naioal College of Busiess Admiisraio ad Ecoomics Gulberg-III, Lahore, Pakisa adakhaariq@gmailcom G M Habibullah Naioal College of Busiess Admiisraio ad Ecoomics Gulberg-III, Lahore, Pakisa musafa1941@yahoocom Absrac I his paper, by usig geeralized hypergeomeric fucios of he ype F, a exesio of he Laguerre polyomials is iroduced ad similar o hose relaig o he Laguerre polyomials, a umber of geeraig fucios ad recurrece relaios for his exeded Laguerre polyomials have bee deermied Mahemaics Subjec Classificaio: 33C45, 5A15, 11B37 Keywords: Polyomials, Geeraig Fucios, Recurrece relaios 1 Iroducio This oe cocers wih he polyomials ( ), A x based o F similar o he Laguerre polyomials deal wih i [4] ad [1] Mos of he classical resuls o he Laguerre polyomials ca be geeralized sraigh away by usig relaio ivolvig hypergeomeric fucios A large umber of releva properies of he Laguerre orhogoal polyomials, is exesios ad is applicaios are available i books ad jourals I his regard we ca refer umerous rece works eg [1], [3], [5], [6], [9], [1], ad [11]

2 19 A Kha ad G M Habibullah There is a wide rage of applicaios of he Laguerre polyomials i may areas icludig i permuaio saisics The momes of he measure for hese polyomials are he geeraig fucios for permuaios accordig o eigh differe saisics Gurlad e al [7] has cosidered a discree disribuio i which he probabiliies are expressible by he Laguerre polyomials, is formulaed i erms of a probabiliy geeraig fucio ivolvig hree parameers May auhors (see [], [8], ad [13] for deails) have sudied problems of permuaio polyomials modulo m, polyomials wih ieger coefficies ha ca iduce bijecios We follow he well kow echiques o describe he properies of he exeded Laguerre polyomials A, ( x ) defied by + 11 A, ( x) = F, ;,1; x, where is ay o-egaive ieger Rewriig i i series form, we have + 1 k k k x A, ( x) = 1 k = () ( k ) 1 k k By direc evaluaio ad usig Lemma 5, pp of [1], we obai k 1 x A, ( x) = k = ( k) ( k) ( k) (11) Also oe ha A k, ( x) 1 x = (1) = = k= ( k) ( k) ( k) which leads o he geeraig fucio A, ( x) 1 x = e F ;,1; (13) = Mai Resuls I his secio, we prove mai resuls ad deermie recurrece relaios for

3 Exeded Laguerre polyomials 191 he exeded Laguerre polyomials A, ( x ) Theorem 1: If c is ay posiive ieger, ad is ay o-egaive ieger he ( c) A, ( x) 1 c c+ 1 1 x = F, ;,1; c = ( 1 ) 1 (1) Proof: From Equaio (1), we oe ha A k, ( x) 1 x ( c) = ( c) ( k) ( k) ( k) = = k= By usig Lemma 11(8), pp 57 of [1], we obai k ( c) A ( ) ( ) k, x + c + k x = = = k= ( k) ( k) k ( c+ k) ( c) k ( x) = k= = ( k) ( k) k 1 ( c) k 1 x = ( 1 ) c k = ( k) ( k) 1, so ha ( c) A, ( x) 1 c c+ 1 1 x = F, ;,1; c = ( 1 ) 1 Wih c = 1, i reduces o 1 x A, ( x) = exp 1 1 () = Theorem : If 1, he Proof: From Equaio (13) ( ) ( ) ( ) xda x = A x A x,,,, 1 d D = (3) dx

4 19 A Kha ad G M Habibullah ( ) 1 A x x, = e F ;,1; = Se A, ( x) σ, ( x) = Suppose ha x 1 x ψ = F ;,1; The x F = eψ = σ, ( x), = (4) provided ha he series is uiformly coverge Parial derivaives of F leads o F F x = F (5) x F F Furhermore ogeher wih = σ, ( x) ad = σ, ( x), x = = Equaio (5), he yields x σ, ( x ) σ, ( x ) = σ, 1 ( x ) = = = 1 I he follows ha σ, ( x) =, ad for > 1, xσ, ( x) σ, ( x) = σ, 1( x) xda ( x) = A ( x) A ( x),,, 1 Theorem 3: If, he DA, ( x) = DA, 1 ( x) DA, ( x) + xa, ( x) (6) Proof: Le F = A()exp x = y, ( x), 1 = (7) F so ha = x A ( )exp x = y, ( x) x 1 1 = (8)

5 Exeded Laguerre polyomials 193 F 1 = x A( )exp x = x F x 1 ( ) Cosequely (9) y, ( x) y, 1 ( x) + y, ( x) = x y, ( x) = = 1 = = I hus follows ha y, ( x) =, y,1 ( x) =, ad for >, ( ) ( ) ( ) ( ) DA x = DA x DA x + xa x Theorem 4: If, he,, 1,, DA ( x) = x ( k 1) A ( x) (1),, k k = Proof: By usig Equaio (8), we obai y, ( x) = x C y, ( x) = = =, k ( ) = x ( k 1) y x = k= Hece, we ge DA ( x) = x ( k 1) A ( x),, k k = Similarly we ca show ha if 3, he A x = ( ) A x (3 4 x ) A x + (3 ) A x ( ) ( ) ( ) ( ),, 3,, 1 Refereces [1] A Akbary, D Ghioca ad Q Wag, O permuaio polyomials of prescribed shape, Fiie Fields ad Their Applicaios, 15(9), [] E Aksoy, A Cesmelioglu, W Meidl ad A Topuzoglu O he Carliz rak of permuaio polyomials, Fiie Fields ad Their Applicaios, 15(9), 48 44

6 194 A Kha ad G M Habibullah [3] S Alam ad A K Chogdar, O geeraig fucios of modified Laguerre polyomials, Rev Real Acad De Ciecias Zaragoza, 6(7), [4] G Adrews, R Askey ad R Roy Special Fucios, Cambridge Uiversiy Press, 1999 [5] K Y Che ad H M Srivasava, A limi relaioship bewee Laguerre ad Hermie polyomials Iegral Trasforms ad Special Fucios, 16(5), 75 8 [6] E H Doha, H M Ahmed ad S I El-Soubhy, Explici formulae for he coefficies of iegraed expasios of Laguerre ad Hermie polyomials ad heir iegrals, Iegral Trasforms ad Special Fucios, (9), [7] J Gurlad, E E Che ad F M Heradez, A ew discree disribuio ivolvig Laguerre polyomials, Commuicaios i Saisics - Theory ad Mehods, 1(1983), [8] I Krasikov ad A Zarkh, Equioscillaory propery of he Laguerre polyomials, J of Approximaio Theory, 16(1), 1 47 [9] D W Lee, Properies of muliple Hermie ad muliple Laguerre polyomials by he geeraig fucio, Iegral Trasforms ad Special Fucios, 18(7), [1] K S Nisar ad M A Kha, A oe o Biomial ad Triomial operaor represeaios of cerai polyomials, I J Mah Aalysis, 5(11), [11] V Radulescu, Rodrigues-ype formulae for Hermie ad Laguerre polyomials, A S Ui Ovidius cosaa, 16(8), [1] E D Raiville, Special Fucios, The Macmilla Compay New York, 196 [13] Q Wag, O iverse permuaio polyomials, Fiie Fields ad Their Applicaios, 15(9), 7 13 Received: Augus, 11