Some Generating Functions for Bessel Function. by Using Lie Theoretic Method

Transcription

1 I.. Coemp. Mh. Scieces Vol. 7 o Some Geerig Fucios or Bessel Fucio Usig Lie Theoreic Mehod D. V. S. Kushwh V. S. Dhr d 3 Rohii Wze 3 Deprme o Mhemics dvs.ushwh@hoo.com Isiue o Iormio Techolog d mgeme Gwlior M.P. Idi 3 Deprme o Mhemics Iegrl Isiue o Iormio Techolog d mgeme Gwlior M.P. Idi src. This pper is emp is mde o oi Geerig ucios o modiied Bessel Fucio. We c id umer o geerig ucios or vrious specil ucio d orhogol polomils he pplicio o group-heoreic mehod iroduced Louis Weiser.The process m lso led o some ew geerig ucios or correspodig specil ucios. Bessel Fucio d orhogol polomils hve specil imporce i egieerig scieces d cosiue good model or m ssems i vrious ields. Kewords: Bessel Fucio Weiser d clidricl ucio ec. INTRODUCTION lier ordir diereil equio coiig prmeer m e wrie i he orm L v...

2 44 D. V. S. Kushwh V. S. Dhr d R. Wze Susiuig or supposig he le memer polomil i we cosruc he pril diereil operor L L o ucio o wo idepede vriles. This operor is idepede o d is commuive wih. soluio o he simuleous equio Lu o u u where v v is cos hs he orm u v where v v is soluio o he u v is soluio o he equio Lu u u. Now suppose h idepedel o he precedig cosiderios we hve oied eplici soluio u g o Lu d h rom he properies o his ucio we ow h i hs epsio i powers o o he orm g Σg Where is o ecessril ieger. I erm wise operor wih L o his series is permissile he L uls ech erm o he series d v g is soluio o. Thus g is geerig ucio or ceri soluios o. The mi prolem is o id g is epsio is deil o clculio. I is diicul i geerl o id eplici soluio o Lu oher h riicil super posiio o he ucios v or which he geerig ucio reduces o uolog. However i he equio dmis group o rsormio esides o i is possile i m cses o id soluio which leds o sigiic geerig ucio o he orm This mehod o oiig geerig ucio is ow s Weiser s mehod. I his chper we emplo Weiser mehod o oi some geerig ucios or Bessel s ucio. GENERTING FUNCTION FOR BESSEL FUNCTION BY WEISNER S METHOD O replcig he prmeer i Bessel s Dierei equio d v dv v... 3 d d he operor he pril diereil equio Lu is cosruced where

3 Geerig ucios or Bessel ucio 44 L.. 4 This operor ulus u v i v sisies 3.. d hece is clidricl ucio o order. Thus ever geerig ucio o se o clidricl ucios is soluio o Lu. I is show h he pril diereil equio Lu is ivri uder hree prmeer Lie group This operor ulus u v i v sisies 3.. d hece is clidricl ucio o order. Thus ever geerig ucio o se o clidricl ucios is soluio o Lu. I is show h he pril diereil equio Lu is ivri uder hree prmeer Lie group. This group is he pplied o he ssemic deermiio o geerig ucios or Bessel s ucio. GROUP OF OPERTORS O he sis o recurrece relio sisied Bessel ucio we cosruc he ollowig hree diereil operors. 3 I c e veriied h hese operors sis he commuor relios [ - ] [ 3 ] [ 3 - ] - - d here ore geere hree prmeer Lie group. From hese relios d he operor idei. - L Where L is he operor 4 i ollows h - 3 re commuive wih - L d hereore cover ever soluio o Lu i o soluio. I priculr

4 44 D. V. S. Kushwh V. S. Dhr d R. Wze Where is rirr comple umer d I order o deermie he muliplier represeio o operors o SL we irs compue he cios o ep ep - To oi ep we eed o iegre he equio d d γ d d γ d dγ d he iiil codiios re γ O solvig hese equios we ge he resulig rsormios deermiig he cio o ep o lic ucio which re give Thereore ep O he oher hd direc epsio we hve ep

5 Geerig ucios or Bessel ucio 443 Equig 3..4 d 3..5 we hve Susio - so we ge. This is well ow geerig ucio. 8 Ph Similrl o oi ep - we eed o iegre he ollowig equios. d d d d γ d he iiil codiios re er solvig hese equios we ge he resulig rsormios deermie he cio o ep - o lic ucio which re give Thereore

6 444 D. V. S. Kushwh V. S. Dhr d R. Wze ep 3..6 O he oher hd direc epsio. ep Equig 3..6 d 3..7 Pu This is well ow Lommel s Secod Formul [ Riville] CONCLUSION This echique clled Weiser s echique is ver powerul echique or oiig geerig ucios. B usig his echique we c id umer o geerig ucios or vrious specil ucio d orhogol polomils i ssemic mer.

7 Geerig ucios or Bessel ucio 445 The process m lso led o some ew geerig ucios or correspodig specil ucios. REFERENCES []. Cherjee S.K 975 Some properies o simple Bessel polomils rom he view poi o lie lger C.R. cd. Bulgre Sci.. []. Erl D. Risville 96 Chelse Pulishig Comp Bro New Yor P. 69. [3]. Gup S. d Bsu O.K. 995 Group heoriic origi o ceri geerig ucios o prolic clidricl ucios C. Mh. Soc. 87 P.7-3 [4]. Iield L. d Hull 95 The corizio mehod Revs. Moder Phs. [5]. Kum B. Mrch 966 Some specil ucio o mhemicl phsics Vol. 7" No. 3 P [6]. Miller W. 964 Oe Lie lgers d some specil ucios o mhemics phsics Memoir No.5 mer Mhs. Soc. [7]. Miller W. r.968 The lie heor d specil ucios New Yor cdemic Press [8]. Ph M Lie heoric geerig ucios o mulivrile geerlized Bessel ucios. [9]. Mooch H. L. 976 Lie heoreic geerig ucios pulicios de [ isiue mhemic Nouvelle series or. []. Weiser L.955 Group heoreic origi o ceri geerig ucios. Pciic. Mh. 5 P []. Weisers Louis 959 Geerig ucios or Bessel ucios CN.. Mh. Vol. PP Received: ugus