Deriving Euler s identity and Kummer s series of Gauss s hypergeometric functions using the symmetries of Wigner d-matrix
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1 Derivig Euler s ietity Kuer s series of Guss s hypergeoetri futios usig the syetries of Wiger -trix Mehi Hge Hss To ite this versio: Mehi Hge Hss. Derivig Euler s ietity Kuer s series of Guss s hypergeoetri futios usig the syetries of Wiger -trix. 07. <hl-09> HAL I: hl-09 Suitte o 8 Mr 07 HAL is ulti-isipliry ope ess rhive for the eposit isseitio of sietifi reserh ouets whether they re pulishe or ot. The ouets y oe fro tehig reserh istitutios i re or ro or fro puli or privte reserh eters. L rhive ouverte pluriisipliire HAL est estiée u épôt et à l iffusio e ouets sietifiques e iveu reherhe puliés ou o ét es étlisseets eseigeet et e reherhe frçis ou étrgers es lortoires pulis ou privés.
2 Derivig Euler s ietity Kuer s series of Guss s hypergeoetri futios usig the syetries of Wiger -trix M. Hge-Hss Uiversité Liise ulté es Siees Setio Hth-Beyrouth Astrt Strtig fro the syetries ofwiger -trix of rottio i the theory of Agulr Moetuwe fi ot oly Euler s ietity ut lsokuer s series of Guss hypergeoetri futios thereltios etwee the eleets of the series.. Itroutio The gulr oetu hs ee stuie for log tie go y y sietists y ifferet ethos [-8] it is prt of the ourse of qutu ehis. But the trix eleets of fiite rottios or Wiger -trix its syetries ws eterie y Wiger the y other sietists y ifferet ethos [-3].These trix eleets re siply expresse i ters of Gusshypergeoetri futios[-3-0] its syetries re esily eterie re oly etioe i the literture. The Guss hypergeoetri futiogh C is solutio of Eulers hypergeoetri ifferetilequtio the preters C re iepeets of.also it is iporttto ote tht Kuer[9] fou -solutios of the hypergeoetri futio equtio re lssifie ito six group s u u orig to the expressio of the preter C. I Shwiger pproh of gulr oetu [] it is esy to opute the Wiger -trix or SU -trix to erive two expressios: the first oe is futio of the vrile =θ the seo is futio of the vrile =osθ oreover the expressios of the syetries of these eleets e erive siply. I this work we will erive the reltios etwee these futios strtig fro the syetries of the trix eleets. We fi siplythe Euler s ietity lso the Kuer s series of GHwe fi tht u = λ i u i λ i = ostt. The pl of this pper is s follows: We egi i setio y reviewig the gulr oetu GH. I setio 3 we erive fro the syetries of SU -trixthe Euler ietity other reltios etwee hypergeoetri futios with C=. I setios to 8 we erive the reltios etwee hypergeoetri futios with C=++- C=+ C=- + C=- fillyc=+-. - Revisio of gulr oetu Hypergeoetri futio
3 The Wiger D-trix or SU D-trixis expresse i ters ofghosequetly we ke reier of these futios. The we follow Shwiger pproh for its sipliity for the lultio of these trix eleets their reltios of syetries. illy we give the expressios of the prtiulr ses with = osθ = si θ = θ = otθ whih re very iportt for the rest of this work..- The hypergeoetri futio[9] is solutio of Eulers hypergeoetri ifferetil equtio: u u - [- ] - u 0.. With. 0! The Pohher syol stisfies x x x x x.3 A x x Arou the poit = 0 two iepeet solutios re It is siple to oserve tht very useful reltioship for the rest of this work..-the Shwiger s relitio of SU i ters of oso opertors [] is: 00 [ i ]. i!! J J J J J With A J x y J 3 x y -The represettio trix of rottios R i J i Jy i J e e e for SU re: i e J i D R e A i Jy os si e u si os i e e i os si e e i i si os The syetries of the D-tries rise fro the uitry of SU the siple expoetilfor of R it s give i ters of SU -trix y:
4 3.3- The speil vlues of the eleets of the rottio SU -trix re: -!!!!! si os. -!!!!! os os si os.7.- A y pplyig the syetry we fi: 3-!!!!! si si si os.8 -!!!!! ot ot si os.9 The uers of reltios etwee hypergeoetri futios re very yso we liit ourselves to those tht re useful for the presettio. 3. The syetries of SU -trix the Eulers ietity Of the hypergeoetri futioswith C=.-Wewrite:.- The reltio os os the Euler s Ietity: We hve:. os si os. os si os
5 With:!!!! After oprig the ove expressios we fi: os os os 3. Let: os A. 3. We eue fro 3. Eulers ietity of hypergeoetri futios: After ietifitio of: si We fi: si os If the os. A 3. We therefore eue the reltioship: 3..- Writig gi si We fi: si si Put: We fi the followig expressio: 3. - The syetries of SU -trix the reltio etwee The hypergeoetri futios with C=++-.- Let us tke: si os A we put p. We fi the reltio:. With! By pplyig the forul[8 p. 9]:
6 x x! [ x!].3 We fi tht:!! Therefore the peruttio of i is well preserve.- we tke: si os. Put si A p. We fi the reltio:.. 3- Let us tke si ot Usig the expressio. we lso fi:.7 os. We oti the se result fro the reltio:.- The futio eig syetri with respet to it follows fro forul.7 tht:.8 - The syetries of SU -trix the reltio etwee The hypergeoetri futioswith C=+ 3.- Let us tke ot Put:. We fi the reltio: 3.! With 3.3! 3.- We tke ot or y pplyig the Euler trsfortio to the expressio. we fi the expressio: We tkes ot si Posig: si. We fi the reltio:
7 We tkes lso os A os..7 We thus oti the reltio: The syetries of SU -trix the reltio etwee The hypergeoetri futioswith C=- + Sie the futios } { re syetri with respet to the syols we eue fro setio five the followig expressios..-.!!. We strt fro ot to fi The syetries of SU -trix the reltio etwee the hypergeoetri futioswith C=-.- We tke: si si A we put 7. 7.!!!! 7.3 We hoose other reltioships: ot ot.- Put: os os A We fi the reltio:
8 Put: si ot os A 7. We fi the reltio: 7..- Put: si or ot os A 7.7 We fi the reltio: The syetries of SU -trix the reltio etwee The hypergeoetri futioswith C=+.- Let us tke: si os Put A os We fi the reltio: 8. With!! 8..- Usig the expressios we fi: Let us tke: os The futio eig syetri with respet to it follows fro forul 8. tht: 8. Colusio: we therefore fi tht syetry is ot restrite to e ut extes to hypergeoetri futios [ ].
9 9. Referees [] Wiger E P 93 Gruppetheorie u ihreaweug uf ie Quteehik Dertospektre» Brushweig: Vieweg [] J. Shwiger O gulr Moetu i qutu Theory of Agulr Moetu E. L.C. Bieehr H. V D New York: Aei Press [3] N. J. Vileki otios spéiles et théorie e l représettio es groupes Duo 99. [ ] A. Messih «Méique Qutique Toes I et II» E. Duo Pris 9 [] M Chihi R. Hgeor Syetries i Qutu Mehis. [] M.E. Rose Eleetry Theory of Agulr Moetu Joh Wiley Sos I. New York 97 [7] A. R. Eos Agulr Moetu i Qutu Mehis Prieto U.P. Prieto N.J. 97 [8] B. Wyoure Clssil Groups for Physiist E. Joh Wiley 973 [9] I. Erelyi Highertrsel futios Vol. M. Grw-Hill New-York 93. [0] I.S. Grshtey I.M. Ryhik tle of itegrls series prouts [] Her Weyl Syetry Prieto Uiversity Press 9 [] M. Hge-Hss Newsiple pplitios of Syetry ivisio lgers reserhgte Jul 0. 8
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