Jourl of Sciece d Ars Yer 4 No 227-6 24 ORIINAL AER ON BILATERAL ENERATIN FUNCTIONS INVOLVIN MODIFIED JACOBI OLYNOMIALS CHANDRA SEKHAR BERA Muscri received: 424; Acceed er: 3524; ublished olie: 3624 Absrc I his oe e hve obied some ovel resuls o bilerl geerig relios ivolvig modified Jcobi olomils b grou-heoreic mehod I fc i secio e hve iroduced lier ril differeil oeror R hich do o seem o hve ered i he erlier ivesigios d he e obied he eeded form of he grou geered b R Fill i secio 2 e hve obied ovel geerig relio ivolvig he olomils uder cosiderio ih he hel of hich e hve roved geerl heorem o bilerl geerig relios of Keords: Bilerl geerig relio Jcobi olomils AMS-2 Clssificio Code: 33C65 INTRODUCTION Secil fucios re he soluios of ide clss of mhemicll d hsicll relev fuciol equios eerig fucios l lrge role i he sud of secil fucios There re vrious mehods of obiig geerig fucios Bu i hs bee foud h grou-heoreic mehod of obiig geerig fucios is much oe oe i comriso o lic mehod The sud of secil fucios i riculr geerig fucios of secil fucios b grou-heoreic mehod s origill iroduced b LWeiser [] hile sudig geerig fucios of Hergeomeric fucio i he er 955 From seveies d ords ie jus fer he ublicio of he moogrh obiig geerig fucios b EBMcBride [2] of he ls ceur Weiser s grou heoreic mehod hs bee uilized b reserchers hile derivig geerig fucios of vrious secil fucios I he rese ricle e hve obied some ovel bilerl geerig relios of modificio of Jcobi olomils b grou-heoreic mehod here is defied b [3]: Derme of Mhemics Bg college O Bg Horh-733 Idi E-mil:chdrsekhrber75@gmilcom ISSN: 844 958
2 ; 2F 2 ; The mi resul of our ivesigio is sed i he form of he folloig heorem For revious orks o bilerl geerig fucios of Jcobi / modified Jcobi olomils oe m refer o he orks [5-9] he here Theorem If here eiss uilerl geerig relio of he form: 2 3 4 σ σ The imorce of he bove heorems lies i he fc h heever oe kos uilerl geerig relio of e 2 he corresodig bilerl geerig relio c oce be rie do from 3 Thus lrge umber of bilerl geerig relios c be obied b ribuig differe vlues o i 2 2 DERIVATION OF THE OERATOR AND ITS EXTENDED FORM OF THE ROU A firs e seek he folloig firs order lier ril differeil oeror: 2 R R R2 R such h 22 R here Ri i 2 re fucios of bu ideede of d is fucio of Noicig he folloig differeil recurrece relio [3]: d 23 d 2 e defie such h 24 R 2 josro
3 25 R We o roceed o fid he eeded form of he grou geered b R ie e shll R fid e f here f is rbirr fucio d is rbirr cos rel or comle If ϕ be soluio of Rϕ d if e rsform he oeror R o E such h E R R2 he ie Therefore e hve E φ R φ Eφ R Rφ φ Eφ e f e f E φ e φ f Fill e choose e vribles X Y so h he oeror E is rsformed io he oeror D Uder his chge of vribles le ϕ f be rsformed X io FXY Therefore b Tlor s heorem e ge R D e f ϕ e FXY ϕ F X Y ϕ g suosig h F X Y is rsformed io g b iverse subsiuio R B he mehod ou-lied e shll comue e f here 2 R Le ϕ be fucio such h Rϕ The o solvig e ge φ Therefore 2 E ϕ R ϕ No le XYbe se of e vribles for hich 26 EX EY so h E reduces o X No solvig 26 e ge se of soluios s follos: X Y From hich e ge XY X Recllig R ϕ Eϕ here ϕ e ge ISSN: 844 958
4 e E R φ E φ e f e f R e [ f ] No he rsformios XY ill rsform E io D X X Mkig he subsiuios d lig he Tlor s heorem e ge E D e f e X Y f XY X D e X Y f X Y X Fill subsiuig X Y e ge f [ ] Therefore e ge f f R R 27 e f e f ALICATION OF THE OERATOR No riig f f f i 27 e ge R 28 e Agi o he oher hd ih he hel of 25 e hve R 29 e R Equig 28 & 29 d uig e ge 2 We o roceed o rove he Theorem b usig he bove geerig relio josro
ISSN: 844 958 5 roof of he Theorem : No he righ hd side of 3 σ [from 4] [from 2] [from 2] Lef hd side of 3 hich is Theorem Fill e ould like o oi i ou h he Theorem c be roved s follos b he direc licio of he oeror R usig he mehod s discussed i [4] We firs cosider he folloig uilerl geerig relio of he form 2 Relcig b i 226 d he oerig R e o boh sides e ge 22 e e R R The lef member of 22 ih he hel of 27 becomes 23 The righ member of 22 ih he hel of 25 becomes 24 Equig 23 d 24 d he uig e ge 25 No relcig b d b d simlifig e ge σ
6 here σ This comlees he roof of he Theorem Ackoledgeme: I m hkful o m suervisor Drrof A K Chogdr Derme of Mhemics IIEST Shibur d m fried Mr Klid Sm for rerig his er REFERENCES [] Weiser L rou heoreic origis of ceri geerig fucios cific J Mh 5 33 955 [2] McBride ED Obiig geerig fucios Sriger-Verlg Ne York Heidelberg Berli 97 [3] Riville ED Secil fucios Mcmill Ne York 96 [4] Chogdr AK Cherje SK BullCl Mh Soc 73 27 98 [5] Chogdr AK Com Fc SciUiv Akr Soc A Mh Sis 4-2 69 994 [6] Chogdr AK Bull Cl Mh Soc 78 363 986 [7] hosh B Some heorems o more geerl geerig fucios of Jcobi olomils di 39-2 55 988 [8] Ds S O ril differeil oerors for Jcobi olomils ure Mh Muscris 3 95 984 [9] Ber CS Chogdr AK Ulr Scieis 24 22 josro