On computing two special cases of Gauss hypergeometric function

Transcription

1 O comuig wo secil cses of Guss hyergeomeric fucio Mohmmd Msjed-Jmei Wolfrm Koef b * Derme of Mhemics K.N.Toosi Uiversiy of Techology P.O.Bo Tehr Ir E-mil: mmjmei@u.c.ir mmjmei@yhoo.com b* Isiue of Mhemics Uiversiy of Kssel Heirich-Ple-Sr. 40 D-4 Kssel Germy E-mil: oef@mhemi.ui-ssel.de Corresodig uhor * Absrc. I his er comuiol echiue is give o re-obi he elici forms of wo cses of he Guss hyergeomeric fucio b c; for b / d c / /. Some secil ideiies reled o he wo foresid cses re lso iroduced. illy secil Mle cge clled ormlpowerseries PS is used o uomiclly comue some resuls give i he er. Keywords. Guss hyergeomeric fucio Symmeric orhogol olyomils Euler ideiy Legedre s dulicio formul Kummer formul. MSC 00: Primry C05 Secodry C90.. Iroducio Le us begi wih he Guss hyergeomeric differeil euio [ 8]: where y β y β y 0 β d re cos rmeers. Sice he idicil euio of i.e. r r 0 hs wo roos 0 r by usig he robeius mehod he series soluio of he Guss euio for r 0 c be eressed s β β β β β β y!!!... where 0... d he series coverges for < <. This Tylor series esio 0 is clled Guss hyergeomeric series d is sum deoed by β or β ; is clled Guss hyergeomeric fucio. So we hve

2 0 0 where! i y i β β. 4 A soluio bsis of differeil euio is give by [] d y y β β 5 i which β d β re ll o-iegers so h he geerl soluio of is give s lier combiio y Ay By where A B re wo coss. The imorce of he Guss hyergeomeric fucio is h my elemery d secil fucios of mhemicl hysics c direcly be eressed i erms of i see e.g. [ ]. Some emles re resecively: ; 6 l ; 7 b d b B b 0 ; 8 rcsi ; 9 rc ; 0 clssicl orhogol olyomils lie he Jcobi olyomils P β β furher s r S r s / ] / [ / ] / [ where ; s r S is he moic form of bsic clss of symmeric orhogol olyomils see [] d s r S r s / / 4 ] / [ ] / [ θ θ θ θ θ θ where ; s r S θ is bsic clss of symmeric orhogol fucios [] h geerlizes he olyomils for θ d filly

3 J ; b c d b cd i d bc c i / d bc 0 b cd i d bc d bc i b cd 4 which is clss of rel olyomils orhogol wih resec o he geerlized T sude b disribuio weigh fucio b c d e rc o c d see [0]. There re lso secil formuls of he Guss hyergeomeric fucio icludig cos rmeers. or isce he Kummer d Guss formuls [8] re wo wellow emles: Γ b Γ b ; b b Kummer formul 5 Γ b Γ b Γ c Γ c b b c; Guss formul. 6 Γ c Γ c b I should be dded h umber of very secil ois he Guss hyergeomeric fucio c ssume secific vlues. or emle i [4 5] i is show h 5 ; ; ;. 4 9 I his reviewig ricle we rese comuiol mehod for re-obiig wo oher secil cses of Guss hyergeomeric fucio i.e. / d / / [.556 Re d Re. 5..0] / i which R i he covergece regio < < d he iroduce some secil ideiies reled o hese wo secil cses. We lso imleme he orml Power Series PS lgorihm which ws desiged d cosruced by W. Koef i 99 [7] i Mle o our comuiol resuls i he e secio.. Comuig / / ; d / / ; Due o Euler s ideiy [ ] e i cos i si i 0

4 d Moivre s ideiy e i e i for R oe ges firs cos i si cos i si i si cos 0 which resuls i cos si 0 0 si cos si cos. The wo bove formuls c be rewrie s cos cos 0 si cos si 0. O he oher hd he coefficies C d C i c be wrie i erms of he Pochhmmer symbol Γ... 4 Γ i which Γ 0 e d urose we should firs oe h deoes he Gmm fucio [8]. or his! Γ. 5!! Γ Γ Now ccordig o Legedre s dulicio formul [] d he wo ideiies Γz z π Γ z Γ z 6 Γ β Γ β Γ β β d Γ β 7 β followig from 4 he deomior erm i he frcio 5 is simlified s 4

5 Γ Γ Γ Γ Γ Γ π 8! Γ Γ Γ. π Therefore we hve π Γ. 9! Γ Γ! Noe i 9 h he Legedre formul imlies π Γ Γ Γ. Coseuely subsiuig 9 io he firs relio of yields: / cos. 0 cos If for simliciy / d ideiy 0 is filly rsformed o / cos rc /. Bu s we oied ou i he covergece regio of he Guss hyergeomeric series is < < d sice rc is jus defied o he rel lie for 0 formul should oly be cosidered for < 0. Therefore we lso comue he elici form of / / ; for osiive 0 <. To rech his gol we use wo differe wys. The firs wy is s follows: b Le z b ib b e i rc d se d b i for 0. So i e i rci rc rci l. By subsiuig his resul i oe ges for 0 / / cos rc cosh l cos i l which is vlid for 0 < d R. The secod wy is o use he Tylor esio 5

6 which subseuely yields / / 0 <. 5 Combiig d 5 gives he followig well-ow ideiy: / / cos rc if if < 0 0 <. 6 or isce for / / 6 yields / / / / 4 cos π. 7 I is ieresig o oice h 6 is lso vlid for he wo boudry ois d becuse ccordig o Guss s formul 6 we hve / Γ/ Γ / π Γ 8 which c direcly be roved for < 0 if oe lies Legedre s dulicio formul. Similrly for ccordig o Kummer s formul 5 we hve 9 / / π Γ / π cos / / Γ Γ / which gi c be roved for < < if oe lies Legedre s dulicio formul d furhermore he well-ow ideiy Γ Γ π si π 0 < <. Similrly o comue / / ; he coefficie C c be rewrie s 40! 6

7 which chges he secod formul of o / si si cos. 4 Now if for simliciy / d 4 becomes / / si rc si rc 4 which holds for < 0 becuse rc is defied o 0. Agi o comue he elici form of / / ; for osiive 0 < here re wo differe wys. The firs wy is o subsiue i 4 i.e. si i l / si rc / si rc i si l sih l sih The secod wy is o ly he Tylor esio l. 4 which resuls i / /. 45 Combiig 4 d 45 gives he secod well-ow ideiy s: / / si rc si rc if if < 0 0 <. 46 or isce if / / 4 he 46 is reduced o 7

8 / / / /. 4 π si 6 47 Moreover for / here eiss limiig cse i 46 which direcly geeres ideiy 0 becuse we hve / si rc rc lim. 48 / / si rc Noe h he resul 46 is lso vlid for he wo boudry ois d becuse ccordig o 6 he ideiy / Γ/ Γ / 49 Γ/ Γ c be roved for < / if oe lies Legedre s dulicio formul d lso ccordig o 5 ideiy / / Γ5 / 4 / Γ/ / Γ5 / 4 / Γ/ π si 50 4 c be roved for < < 5 if Legedre s dulicio formul d he relio Γ Γ π si π re used simuleously... Some ideiies reled o / / ; d / / ; Sice relios 0 d 4 hve rigoomeric forms vrious ideiies c be derived for he wo meioed hyergeomeric fucios. Here we iroduce hree emles of hese ideiies. i By referrig o 0 d 4 oe c firs coclude / si / si /. / cos 5 Clerly his relio is simlified s follows if oe ssumes / d : 8

9 9 / / / / / /. 5 ii Sice we hve / cos cos cos / 5 by ig d i 5 d oig he well-ow ideiy [] / / rc cos 54 we filly obi / / / / / / /. 55 iii Sice / cos cos si si / 56 gi ig d i 56 yields / / / / / / / Aedi: PS Algorihm yields revious resuls I 99 W. Koef iroduced he ormlpowerseries PS lgorihm [7] d imlemeed i i he Mle sofwre. The lgorihm is ccessible vi he Mle commd cover ormlpowerseries see lso [4]. I his secio usig his secil cge we wish o direcly comue some resuls give i he er d oi ou h hey c be deermied comleely uomiclly from righ o lef by he PS lgorihm see lso [9]. Similrly oe h he lgorihmic rocedure give by Roch i [] which is lso rilly imlemeed i Mle yields he coversio from lef o righ.

10 .. Mle comuios > red "hsum.ml": Pcge "Hyergeomeric Summio" Mle V - Mle Coyrigh Wolfrm Koef Uiversiy of Kssel ormul 6 : > fs:cover-^ps; fs : 0 > hyer:sumohyerofs; ochhmmer! hyer : Hyergeom [ ] [ ] > simlifysubshyergeomhyergeomhyer; ormul 9: > fs:coverrcsisr/srps; fs : 0 > hyer:sumohyerofs;! 4! hyer : Hyergeom > simlifysubshyergeomhyergeomhyer; rcsi ormul 0: > fs:coverrcsr/srps; fs : - 0 > hyer:sumohyerofs; hyer : Hyergeom > simlifysubshyergeomhyergeomhyer; 0

11 l ormul : > fs:coverjcobiplhbeps; fs : 0 biomil - ochhmmer ochhmmer β / ochhmmer! > hyer:coversumohyerofsbiomil; hyer : Hyergeom [ β ] [ ] biomil ormul 0: > fs:covercos*rc/cosrc^ps; fs : - ochhmmer! 0 > hyer:sumohyerofs; hyer : Hyergeom > simlifysubshyergeomhyergeomhyer; 4 / I 4 / I ormul : > fs:cover-^-*cos**rcsr-ps; fs : 0 > hyer:sumohyerofs; ochhmmer! 4 / hyer : Hyergeom > simlifysubshyergeomhyergeomhyer;

12 4 / cos rcsi 4 / / 4 / ormul 5: > fs:cover/*sr^-*-sr^-* PS; fs : 0 > hyer:sumohyerofs; ochhmmer! hyer : Hyergeom > simlifysubshyergeomhyergeomhyer; 4 / cos rcsi 4 / / 4 / ormul 46: > fs:cover-^-*si*-*rcsr-/ *-*sircsr-ps; fs : 0 > hyer:sumohyerofs; ochhmmer! hyer : Hyergeom > simlifysubshyergeomhyergeomhyer; 4 / 4 / si rcsi 4 /

13 Refereces []. Abrmowiz M. d Segu I.A. 97 Hdboo of Mhemicl ucios wih ormuls Grhs d Mhemicl Tbles 9h ed NewYor: Dover. []. Arfe G. "Hyergeomeric ucios.".5 i Mhemicl Mehods for Physiciss rd ed. Orldo L: Acdemic Press []. Bres E. W. A ew develome i he heory of he hyergeomeric fucios. Proc. Lodo Mh. Soc [4]. Geddes K. Lbh G. Mog M.: Mle dvced rogrmmig guide. Mlesof 008. [5]. Gessel I. d So D. Srge evluios of hyergeomeric series. SIAM J. Mh. Al [6]. Goser R. W. Decisio rocedures for idefiie hyergeomeric summio. Proc. N. Acd. Sci. USA [7]. Koef W. Power series i comuer lgebr. J. Symbolic Comuio [8]. Koef W. Hyergeomeric Summio Vieweg Bruschweig / Wiesbde 988. [9]. Koef W. Comuerlgebr. Sriger Berli 006 [0]. Koef W. Msjed-Jmei M. A geeric olyomil soluio for he differeil euio of hyergeomeric ye d si seueces of clssicl orhogol olyomils reled o i Iegrl Trsforms Sec. uc []. Msjed-Jmei M. A bsic clss of symmeric orhogol fucios usig he eeded Surm-Liouville heorem for symmeric fucios J. Comu. Al. Mh []. Msjed-Jmei M. A bsic clss of symmeric orhogol olyomils usig he eeded Surm Liouville heorem for symmeric fucios J. Mh. Al. Al []. Roch K. Hyergeomeric fucio rereseios. I: Proceedigs of he Ieriol Symosium o Symbolic d Algebric Comuio [4]. Zucer I. J. d Joyce G. S. Secil vlues of he hyergeomeric series III. Mh. Proc. Cmbridge Philos. Soc [5]. Zucer I. J. d Joyce G. S. Secil vlues of he hyergeomeric series II. Mh. Proc. Cmbridge Philos. Soc