We prove some new formulae for the derivatives of the generalized Gegenbauer
polynomials associated to the Lie algebra $A_2$.Comment: 3 pages, no figures; submitted to Theor. Math. Phy

Fuertes, W. Garcia

Perelomov, A. M.

English

arXiv

Preprint.[EN]We prove some new formulae for the derivatives of the generalized Gegenbauer polynomials associated to the Lie algebra $A_2$.[ES]Probamos algunas nuevas fórmulas para las derivadas de los polinomios generalizados Gegenbauer asociados al álgebra de Lie $ A_2 $

García Fuertes, Wifredo

Perelomov, Askold M.

Gestion del Repositorio Documental de la Universidad de Salamanca

arXiv:math-ph/0201026v1  14 Jan 2002On thederivativesofgeneralized GegenbauerpolynomialsW .Garca Fuertes,A.M .Perelom ovDepartam ento de Fsica,Facultad de Ciencias,Universidad de Oviedo,E-33007 Oviedo,SpainA bstractW e prove som e new form ulae for the derivatives of the generalized G egenbauerpolynom ialsassociated to the Lie algebra A 2.Asitiswellknown[1],theclassicalGegenbauerpolynom ialsC m(z)suer,whendierentiatedin z,a shiftin theparam eter,nam elydP mdz= m P+ 1m   1;Pm(z) =m !()mCm(z2); ()m = ( + 1):::( + m   1)TheclassicalGegenbauerpolynom ialsare(up to a factor)theeigenfunctionsofthesim plestquantum Calogero-Sutherland Ham iltonian [2],[3],thatrelated to the Lie algebra A 1. Itisthe purpose ofthisnote to show thatthe sam e shiftin  takesplace in the derivatives ofthegeneralized Gegenbauerpolynom ialsP m ;n(z1;z2)givingthequantum eigenfuctionsoftheCalogero-Sutherland system with Liealgebra A 2:Pm ;n= "m ;n()Pm ;n(z1;z2);Pm ;n= zm1zn2+ lowerterm s;= (z21  3z2)@2z1+ (z22  3z1)@2z2+ (z1z2   9)@z1@z2 + (3 + 1)(z1@z1 + z2@z2)"m ;n() = m2+ n2+ m n + 3(m + n);see[4],[5],[6],[7],[8].Specically,wewillprovethefollowing form ulae:@P m ;n@z1= m P+ 1m   1;n+ A m ;n()P+ 1m   2;n  1+ B m ;n()P+ 1m ;n  2(1)@P m ;n@z2= nP+ 1m ;n  1+ A n;m ()P+ 1m   1;n  2+ B n;m ()P+ 1m   2;n; (2)O n leave ofabsence from the Institute for Theoreticaland Experim entalPhysics,117259,M oscow,Russia.CurrentE-m ailaddress:perelom o@ dftuz.unizar.es1whereA m ;n() =m (m   1)n(m + n +    1)(m + n + )(m +    1)(m + )(m + n + 2   1)(m + n + 2)B m ;n() =  n(n   1)(m + n + )(n +    1)(n + ): (3)Considerrst(1).Theproofofthisform ula proceedsby induction on thesecond quantumnum ber.Thegenerating function fortheJack polynom ialsP m ;0isknown to be[9](1  z1t+ z2t2   t3)   =1Xm = 0()mm !Pm ;0tm: (4)Dierentiation ofthisexpression showsthe validity of(1)when n = 0.On theotherhand,wecan usetherecurrencerelationsforthegeneralized Gegenbauerpolynom ials[7]toexpressP m ;nin term sofpolynom ialswith lowern:Pm ;n= z2Pm ;n  1   ~am ;n  1()Pm   1;n  1  cn  1()Pm + 1;n  2(5)with~am ;n() =m (n + m + )(m   1+ 2)(n + m   1+ 3)(m + )(m   1+ )(n + m + 2)(n + m   1+ 2);cn() =n(n   1+ 2)(n + )(n   1+ ): (6)Dierentiating(5)with respecttoz1 undertheassum ption that(1)isvalid when thesecondquantum num berislowerthan n,and applying therecurrence relation (5)to getrid oftherem aining z2 factors,weobtain:@P m ;n@z1= m P+ 1m   1;n+ [A m ;n  1()~am   2;n  2( + 1)  A m   1;n  1()~am ;n  1()]P+ 1m   3;n  2+ [A m ;n  1()+ m ~am   1;n  1( + 1)  (m   1)~am ;n  1()]P+ 1m   2;n  1+ [B m ;n  1()  (m + 1)cn  1()+ m cn  1( + 1)]P+ 1m ;n  2+ [B m ;n  1()cn  3( + 1)  B m + 1;n  2()cn  1()]P+ 1m + 1;n  4+ [  ~am ;n  1()B m   1;n  1()+ ~am ;n  3( + 1)B m ;n  1()+ A m ;n  1()cn  2( + 1)  A m + 1;n  2()cn  1()]P+ 1m   1;n  3(7)and by explicituseof(3)and (6),wend:A m ;n  1()~am   2;n  2( + 1)  A m   1;n  1()~am ;n  1() = 0B m ;n  1()cn  3( + 1)  B m + 1;n  2()cn  1() = 0  ~am ;n  1()B m   1;n  1()+ ~am ;n  3( + 1)B m ;n  1()+A m ;n  1()cn  2( + 1)  A m + 1;n  2()cn  1() = 0 (8)2andA m ;n  1()+ m ~am   1;n  1( + 1)  (m   1)~am ;n  1() = A m ;n()B m ;n  1()  (m + 1)cn  1()+ m cn  1( + 1) = B m ;n() (9)which establishesthedesired result.Theproofof(2)takesadvantageofthetwin recurrencerelation to (5),see[7],and iscom pletely analogous.In conclusion wewould liketo m entionthattheapproach ofthisnotem ay beused also fortheA n case.W ehopeto return to thisproblem in thefuture.A cnow ledgm entsW e are gratefulto Prof. M .Lorente for interesting discussions. One ofthe authors (A.M .P.)would liketo expresshisgratitudeto theDepartm entofPhysicsoftheUniversity ofOviedo forthehospitality during hisstay asa Visiting Professor.R eferences[1]Batem an H.and ErdelyiA.,Higher TranscendentalFunctions,Vol.II,M cGraw-HillBook Co.,1953.[2]Calogero F.,J.M ath.Phys.12,419{436,1971.[3]Sutherland B.,Phys.Rev.A 4,2019{2021,1972.[4]Olshanetsky M .A.and Perelom ov A.M .,Phys.Rep.94,313{404,1983.[5]Perelom ov A.M .,J.Phys.A 31,L31{L37,1998.[6]Perelom ov A.M .,Ragoucy E.and Zaugg Ph.,J.Phys.A 31,L559{L565,1998.[7]Perelom ov A.M .,J.Phys.A 32,8563{8576,1999.[8]Perelom ov A.M .,in Proc.ClausthalConference1999,LieTheory and ItsApplica-tionsin PhysicsIII,139{154,W orld Scientic,Singapore,2000.[9]Jack H.,Proc.Roy.Soc.Edinb.A 69,1{18,1970.3

On the derivatives of generalized Gegenbauer polynomials

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Gestion del Repositorio Documental de la Universidad de Salamanca