On the Rodrigues Formula Solution. of the Hypergeometric-Type Differential Equation

Transcription

1 Itrtiol Mthticl Foru Vol. 8 3 o HIKARI Lt.-hikri.co htt://x.oi.org/.988/if O th Rorigus Forul Solutio of th Hrgotric-T Diffrtil Eutio W. Roi Egirig Mthtics Grou Eiurgh Nir Uirsit olito Ro EH 5DT UK.roi@ir.c.uk oright 3 W. Roi. This is o ccss rticl istriut ur th rti oos Attriutio Lics hich rits urstrict us istriutio rrouctio i iu roi th origil ork is rorl cit. Astrct I this r rst ssttic roch to th solutio of th hrgotric-lik iffrtil utio its ssocit utio. Th tho roucs tout court th grl solutio of ths utios i th for of coitio of str Rorigus forul grli Rorigus forul of t u origill to Goçls 5 rctl cosir gi Ar t l. I itio ol lsis of clss of itgrls triig th grli Rorigus forul is gi hich colts origil lsis of Ar t l. Fill th rltio t th hrgotric-t iffrtil utio th hrgotric utio is lucit furthr folloig th ork of Kof Msj-Jl 7. Mthtics Sujct lssifictio: 335; 3345 Kors: hrgotric-t iffrtil utio; Rorigus forul; grli Rorigus forul; itgrtio tchiu. Itrouctio osir th hrgotric-t sco-orr lir orir iffrtil utio ith o-gti itgr

2 456 W. Roi λ hr is urtic fuctio lir fuctio λ is it of. As usul th shs ot iffrtitio ith rsct to th fuctio rgut i this cs th ril. Our itrst hr is i xtrctig ssttic rocss Rorigus forul solutio to utio. Th tho is isir ol r Goçls 5 lthough i th currt ork r l to liit rticulr ssutios Goçls for xl tht th λ of utio tks th rticulr for tht ri lo. Th tho tht ot hs thr rts. First is rritt i slf-joit for i th usul 8 th trsfor ito th forl joit of utio. Nxt ssu th solutio of th forl joit of 6 lo is gi s th th-riti of othr sco-orr lir orir iffrtil utio of ffctil th s clss s ut this ti of rticulr ihoogous for. c Fill ssu tht this sco iffrtil utio is xct itgrt irctl. Th outco of th o thr-rt rci is sst of four utios i four ukos tht rg fro th tho hich h sol coi ith th rst of th lsis ls to th grl solutio of hich hs s scil cs Rorigus forul fort. I fct s c o rious ork r l to ri th grl solutio to utio ith th sco solutio risig utoticll fro th grl tho. This sco solutio is gi i trs of grli Rorigus forul cosir rctl Ar t l though ri lr Goçls 5. Furthr r l to l th tho to th so-cll ssocit utio 6 of utio utio 4 lo oti th grl solutio to 4 s coitio lso of Rorigus forul grli Rorigus forul. Wh cosirig th grli Rorigus forul it ros cssr to tri rticulr clss of itgrls. This rol s stui i til Ar t l ut thr is g i thir iscussio hich fill-i hr rlsig th rol of lutig th si itgrls i tirl iffrt r. Th coctio ith th hrgotric utio is lso iscuss.. Th Solutio Mtho First is rritt i slf-joit for s 8 λ hr th solutio of th rso iffrtil utio

3 Rorigus forul is gi 4 Folloig Goçls 5 trsfor utio th chg of t ril 5 so tht utio cos λ 6 ith utio 6 ig th forl joit of utio. Eutio 6 is th strtig oit lso of th lsis of Ar t l. Hor th tils of th lsis rst hr iffrs fro tht of Goçls 5 Ar t l. Nxt ssocit utio 6 ith ihoogous sco-orr lir orir iffrtil utio of th s clss. W rit this utio ith t ril s r μ 7 ith r lir fuctio μ it of to tri ritrr oloil of gr. W o ssu tht th th riti of 7 rucs to 6 xctl. This gis ris to th ititis r μ λ 8 9 Fill th lst rt of th tho ssus tht utio 7 is xct i tht rit folloig 5 gi r l μ ith l to tri.

4 458 W. Roi Th ititis 8 l th itifictio of th four ukos: λ μ l r. Usig Liit rul for th th-riti fi tht th itit 8 ils r μ r λ hil forcig th itit ls to l r So fro utios h l μ r 3 l 3 μ 3c λ 3 3. Th Solutios W c o strt to roll-ck th o rocss solig 9 for th sustitut i 9 for th sustitut i 5 for. First rrit usig 3 to gt 4 Itgrtig utio 4 rtl gt ith D 5 D for ritrr itgrtio costts ritrr

5 Rorigus forul 459 oloil of gr. Nxt sustitut 5 ck through 9 5 to gt th grl solutio of utio s s 5 gi D 6 If st D i 6 th gt rticulr solutio to s 7 7 is i th Rorigus forul solutio to sujct to th coitio 3. Th ctul fors tht th r rst for xl i rfrc. Itrstigl if st i 6 th gt sco rticulr solutio to s D 8 8 is th xt Rorigus forul solutio to iscuss rctl Ar t l. Solutio 8 is sujct to th coitio 3 lso. 4. Th Associt Eutio W cosir o th rltioshi 6 t th sic utio its socll ssocit utio ith to roucig Rorigus forul solutio for th ssocit utio. If iffrtit tis fi tht ith gt hr λ 9 λ λ Artl utio 9 ith i i is of th s clss s utio so tht utio 9 cos ithi th coss of th Rorigus forul solutio thoolog c rit o Rorigus forul solutio to utio 9. I sic is oloil of gr hs

6 46 W. Roi Rorigus forul solutio of th for 6 ut ith lloc for th iffrc i gr of th oloils th scific for of th cofficits. λ Mkig ths llocs fi tht th Rorigus forul for is D ith D it of hr is solutio of so tht 3 Th ssocit utio is oti fro 9 trsforig 9 ito th s fort s tht is ish to rit 9 s λ 4 ith λ to tri. I fct it is ll-ko tht 6 5 λ λ λ 6 Rltios 4 6 il th usul xrssios for th ssocit utio. Fill fro 3 5 gt th Rorigus forul solutio to 4 th ssocit utio s D 7 Sttig D i 7 gt or str Rorigus forul

7 Rorigus forul 46 solutio to 4 s 8 S Tl for so xls to ithi cotiol orliig costt. Whil sttig i 7 gt sco t of Rorigus forul solutio to 4 s D 9 Tl. Iortt Associt ODE thir Rorigus Forul ith c costt 5. Th Itgrls Th itgrls Associt Eutio Eutio For Rorigus Forul Lgr Hrit H H H Grli Lgurr L L L L Bssl J J J J Jcoi / /

8 46 W. Roi I 3 h stui i til Goçls 5 Ar t l. Hor th lsis rst i this sctio shoul s s colt to th ork of Ar t l. Eltr iffrtitio shos tht th I stisf th iffrtil utio or I I 3 I I I I 3 hich suggst lookig for risig lorig ortors ith th it of K' s K I I 33 K I I 33 hr h ssu tht 3 c ritt s K K I I 34 ith Tkig 33 cos 35 K 35 K K 35c c 36 c K I I 37

9 Rorigus forul 463 hich is hr Ar t l 3 gi thir lsis of I ; so tk s ko. Sustitutig ck gt c K K fro 35 35c. 5. Discussio oclusios I this fil sctio ill iscuss th rltio of our thoolog to rlt ork ith rticulr hsis gi to coriso ith th root r of Goçls 5 hich isir th currt roch lso to th ork of Ar t l to hich th rsults of th currt uir r lso closl rlt. Also for coltss ill lucit th coctio t th hrgotriclik utios iscuss o th hrgotric utio itslf. First ith rgr to Goçls r 5 ot tht hil th currt tho hs lo ftr clos stu of this ork Goçls roch s tht of folloig rticulr th tors his gol h i ot st-out th grl rocur rst hr. I usig th thoolog st-out i sctio h ri th xrssios 3 for th iffrtil utios rtrs ssuig ol th grl fors 7 hrs Goçls roch ssu ths rsults xlicitl or ilicitl. Furthr Goçls 5 os ot iscuss th ssocit utio 4 lthough h os iscuss th ritis of 6 ut os ot cosir th grl cs iscuss i sctio 3. W ot i ssig though tht Jfrih Fkhri 6 iscuss th ssocit utio uot ltrti for of th Rorigus forul 8; lso scil css of 8 r rst i hghlou Fkhri 34. Fill Goçls trtt of th clss of itgrls rrst rltios 3 is r scific soht iffrt rol tht tht trt hr Arl t l is cosir 5. Th thoolog ig so scific is iffrt too. Nxt turig to th rct ork of Ar t l ot tht s ll s loig thir roch to utio i iffrt r to ithr Goçls 5 or th currt r th h to utili ritio of rtrs to rouc th grli Rorigus forul hrs i th currt roch s i Goçls 5 origil ork th full grl solutio rs t oc fro th sic thoolog. This is so lso for th ssocit utio utio 4 s ll s th origil utio. As to th itgrls I rig i th grli Rorigus forul th lsis of sctio 4 los thir solutio thoolog iitio ith th rsults o-tilig ito th ork of Ar t l thus closig g i thir iscussio coltig th lsis of th

10 464 W. Roi I. All tils of th clcultios folloig utio 37 r coti i Ar t l xct for th tritio of K hich is ltr is oitt. Moig o ill o lucit th rltioshi t utios 9 4 th hrgotric utio for costts α β γ x x x α β x γ x αβ x 38 First cosir utio rit-out λ xlicitl s λ 39 ith th k ' s k ' s costts. Nxt i folloig 7 k th lir sustitutio x ith costts to trsfor ito x x x roi x ± 4 4 x x 4 4 ± 4 4 Th corig s tht 4 is scil cs of 38 ith α ± 4 β γ 4 4 It follos o 7 tht th solutio to th hrgotric-t iffrtil utio c ritt i trs of th hrgotric utio 38. B utiliig th o chg of it ril c xt th thoolog of 7 lucit th rltioshi t utio 9 th hrgotric utio. First rit-out th λ xlicitl s λ 43

11 Rorigus forul 465 Nxt i 9 follo 7 gi k th lir sustitutio x to trsfor 9 ito x x x x ± 4 4 x x 44 roi r tri 4 s for. Th corig s tht 44 is scil cs of 38 ith α β 45 ± 4 γ 45 4 It follos o tht th solutio to th hrgotric-t iffrtil utio 9 c ritt i trs of th hrgotric utio 38. Fill h th solutio of th ssocit utio 4 i trs of tht of th hrgotric utio s ko lr tht 46 Nturll thr r tils i th ritig-out of th rltioshis t th rious utios h cosir th hrgotric utio 7 ut h fiish th iortt rt of th rol th rst of th rol is ttr of ottio. For xl th lot of th solutio of utio i trs of solutio of th utio 38 is rst i til i 7. I sur ssttic roch to th solutio of th hrgotriclik iffrtil utio its ssocit utio hs lo. Furthr ol lsis of clss of itgrls triig th grli Rorigus forul risig through th solutio thoolog hs gi th rltioshis t utios 9 4 utio 38 rst.

12 466 W. Roi Rfrcs Ar I. Goo E. Roux A. Zro A.: Hrgotric-t iffrtil utios: sco ki solutios rlt itgrls. Jourl of outtiol Ali Mthtics Bukrs F.: Guss Hrgotric Fuctio. rogrss i Mthtics hghlou A. Fkhri H.: Surstr Arochs to th Ril Bou Stts of th Hrog-lik Atos. Itrtiol Jourl of Qutu histr H. Fkhri H. hghlou A.: Lr ortors rcursio rltios for th ssocit Bssl oloils. hsics Lttrs A Goçls J. V.: Sur L Forul Rogrigus. ortugli Mthtic Jfrih M. A. Fkhri H.: Surstr sh iric i iffrtil utios of thticl hsics. hsics Lttrs A Kof W Msj-Jl M.: A gric oloil solutio for th iffrtil utio of hrgotric t six sucs of orthogol oloils rlt to it. Itgrl Trsfors Scil Fuctios Nikiforo A. F. Uro V. B.: Scil Fuctios of Mthticl hsics. Birkhusr 988. Rci: Jul 7 3