An Extension of Hermite Polynomials

Transcription

1 I J Coemp Mh Scieces, Vol 9, 014, o 10, HIKARI Ld, wwwm-hikricom hp://dxdoiorg/101988/ijcms A Exesio of Hermie Polyomils Ghulm Frid Globl Isiue Lhore New Grde Tow, Lhore, Pkis G M Hbibullh Globl Isiue Lhore New Grde Tow, Lhore, Pkis Copyrigh 014 Ghulm Frid d G M Hbibullh This is ope ccess ricle disribued uder he Creive Commos Aribuio Licese, which permis uresriced use, disribuio, d reproducio i y medium, provided he origil work is properly cied Absrc I his pper, we iroduce exeded Hermie polyomil H ) ( x ), we esblish is differe forms, geerig fucios, bsic recurrece relios, he pure recurrece relio, he secod order differeil equio, Rodrigues formul d orhogoliy of he exeded polyomil Mhemics Subjec Clssificio: 33C45, 11B37, 05A15 Keywords: Hermie polyomil, exeded Hermie polyomil, recurrece relio, geerig fucio, Rodrigues formul, orhogoliy 1 Iroducio Hermie polyomils re oe of he mos sigific clssicl orhogol polyomils rise from he expressios of he form exp( x ) H( x) 0! Lrge dediced lierure is vilble o sudy he orhogol polyomils We refer some of hem begiig wih [1] followed by [10] d [] We lso refer some rece work regrdig properies, exesios, geerlizios d pplicios of Hermie d he reled orhogol polyomils; eg [5], [3], [7], [6], [4], [9] d [8]

2 456 Ghulm Frid d G M Hbibullh For y o-zero rel umbers d y o-egive rel umber b, we defie geerlized Hermie s polyomil H ) ( x ) s ) exp( x b ) H ( x), (11) 0! for ll fiie x d I follows from (11) h k ) ( x) ( b ) H ( x) 0! 0! k0 k! Usig Lemm 11, pp 57 of [1], we hve k k (, b) ( b) ( x) 0! 0 k0 k!( k)! H ( x) I follows, by compriso, h k k ) ( b) ( x) H ( x)!, (1) k0 k!( k)! which is polyomil of degree precisely i x d (, b) H ( x) x ( x), where ( ) x is polyomil of degree i x Moreover, (, b) (, b) H ( x) ( 1) H ( x) Mi Resuls Sice he proofs of re rdiiol orieed, we lis he properies relig he exeded polyomil The properies ivolve differe forms, geerig fucios, recurrece relios, Rodrigues formul d orhogoliy for he exeded Hermie polyomil If 0 d b 0, (, b) (, b) (, b) x H ( x) b H ( x) H ( x), (1) 1 H ( x) H ( x), () (, b) (, b) 1 for ech m sisfyig 1m, we hve ( m ) H ( x) ( 1)( )( m 1) H ( x), (3) (, b) m (, b) m H ( x) x H ( x) b( 1) H ( x), (4) (, b) (, b) (, b) 1, (5) (, ) (, ) (, ) b H b ( ) b ( ) b x x H x H ( x) 0

3 A exesio of Hermie polyomils 457 (, b) d H ( x) ( b) exp( x )( ) exp( x ) x, (6) d( ) b (, b) (, b) exp( x ) H ( x ) H m ( x ) dx 0, m, (7) 1 (, b) 1 b x H x dx exp( ) ( )!, (8) (, b) 1 H ( x) ( x) F0(, ; ; ), x (9) (, b) c c c 1 ( c) H ( x) (1 x) F0(, ; ; )! (1 x) (10) 0 Jus o illusre he rgumes used here i, we prove he resuls (6) d (7) Theorem 1: (Rodrigues formul) The exeded Hermie polyomil H ) ( x ) is give by (, b) b d H ( x) ( ) exp( x ) exp( x ) dx ) Proof: Le f ( x, ) exp( x b ) H ( x) 0! I follows by Mcluri's heorem h ) f ( x, ) ( ), H x 0! 0 0! which implies h ) H ( x) exp( x ) exp( ( x b) ) 4 b b 0 Subsiuig w x b, we obi b exp( ( ) ) ( ) exp( ) x b b w b w Thus, i follows by (11) h (, b) d H ( x) ( b) exp( x )( ) exp( x ) x, d( ) b or lerively, (11)

4 458 Ghulm Frid d G M Hbibullh (, b) b d H ( x) ( b) exp( x ) exp( x ) ( b) H ( x) dx Theorem : (Orhogoliy of H ) ( x )) (, The simple se H b) ( x) sisfies x H (, b) (, b) x H m x dx m exp( ) ( ) ( ) 0, x Proof: Muliplyig (5) by exp( ), we hve (, b) (, b) exp( x ) H ( x) exp( x ) H ( x) 0 (1) b Similrly, we c wrie (, b) m (, b) exp( x ) Hm ( x) exp( x ) Hm ( x) 0 (13) b Muliplyig (1) by ) Hm ( x ) d (13) by H ) ( x ) d he subrcig he resulig relios, we hve (, b) (, b) ( m )exp( x ) H ( x) H m ( x) exp( x ) p( x), b (, b) (, b) (, b) (, b) p( x) H ( x) H ( x) H ( x) H ( x) where m m I follows h (1) () Refereces x H (, b) (, b) x H m x dx m exp( ) ( ) ( ) 0, [1] E D Riville, Specil Fucios, The Mcmill Compy, New York, 1960 [] G Adrews, R Askey d R Roy, Specil Fucios, Cmbridge Uiversiy Press, 004 [3] J L Lopez, Hermie Polyomil i Asympoic Represeios of Geerlized Beroulli, Euler, Bessel d Buchholz Polyomils, J Mh Al Appl, 39(1999), [4] M Powiersk, O he re of covergece of some orhogol polyomil expsios, J Iequl Pure Appl Mh, 9(008), 9-11 [5] M Rosler, Geerlized Hermie Polyomils d he He Equio for Dukl Operors, Commu Mh Phys, 19(1998),

5 A exesio of Hermie polyomils 459 [6] R Diz, d E Prigu, O hypergeomeric fucios d Pochhmmer k - symbol, Divulg M, 15(007), [7] S B Trickovic d M S Skovic, O he orhogoliy of clssicl orhogol polyomils, Iegrl Trsform Spec Fuc, 14(003), [8] S Mubee, A iegrl represeio of some k -hypergeomeric fucios, I Mh Forum, 7(01), [9] V Rdulescu, Rodrigues-ype formule for Hermie d Lguerre polyomils, A S Uiv Ovidius Cos, 16(008), [10] Z X Wg d D R Guo, Specil Fucios, World Scieific Publishig Co Pv Ld,1989 Received: Jue 15, 014