A matrix Rodrigues formula for classical orthogona

arXiv:math/0610128v1[math.CA]3Oct2006AMATRIXRODRIGUESFORMULAFORCLASSICALORTHOGONALPOLYNOMIALSINTWOVARIABLESMAR´IA´ALVAREZDEMORALES,LIDIAFERN´ANDEZ,TERESAE.P´EREZ,ANDMIGUELA.PI˜NARAbstract.Classicalorthogonalpolynomialsinonevariablecanbecharac-terizedastheonlyorthogonalpolynomialssatisfyingaRodriguesformula.Inthispaper,usingthesecondkindKroneckerpowerofamatrix,aRodriguesformulaisintroducedforclassicalorthogonalpolynomialsintwovariables.1.IntroductionOneofthemostimportantcharacterizationsforclassicalorthogonalpolynomialsinonevariable(Hermite,Laguerre,JacobiandBessel)istheso–calledRodriguesformula(see,forinstance,[3]).Usingthiskindofformulawecanwritethen–thclassicalorthogonalpolynomialintermsofan–thorderderivative.Infact,ifwedenoteby{Pn}naclassicalfamilyoforthogonalpolynomialsinonevariable,then(1)Pn(x)=knω(x)dndxn(φ(x)nω(x)),n=0,1,2,...,whereknisaconstant,φ(x)isapolynomialofdegreelessthanorequalto2,independentofn,andω(x)isanintegrablefunctioninaappropriatesupportset.Ifdegφ=0,Hermitepolynomialsappear,uptoalinearchangeinthevariable.Ifdegφ=1,Laguerrepolynomialsareobtained,andifdegφ=2,wecandeducetwofamiliesofpolynomials,Jacobipolynomialswhenφ(x)hastwosimpleroots,andBesselpolynomialswhenφ(x)hasadoubleroot.Formula(1)iscalledRodriguesformula,honoringB.O.Rodrigueswhoestab-lishedtheformulain1814forLegendrepolynomials.OrthogonalpolynomialsintwovariableswhicharesolutionsofpartialdifferentialequationsweresystematicallystudiedbyH.L.KrallandI.M.Sheffer([14]),in1967.Theydefinedclassicalorthogonalpolynomialsintwovariablesasthesequencesoforthogonalpolynomials{Ph,k}h,k≥0suchthateverypolynomialPh,k,withh+k=n,satisfiesthesecondorderPDE(2)L[w]≡awxx+2bwxy+cwyy+dwx+ewy=λnw,wherea(x,y)=ax2+d1x+e1y+f1;b(x,y)=axy+d2x+e2y+f2;c(x,y)=ay2+d3x+e3y+f3;d(x,y)=gx+h1;e(x,y)=gy+h2,andλn=an(n−1)+gn.2000MathematicsSubjectClassification.42C05;33C50.Keywordsandphrases.Orthogonalpolynomialsintwovariables,classicalorthogonalpoly-nomials,Rodriguesformula.PartiallysupportedbyMinisteriodeCienciayTecnolog´ıa(MCYT)ofSpainandbytheEu-ropeanRegionalDevelopmentFund(ERDF)throughthegrantMTM2005–08648–C02–02,andJuntadeAndaluc´ıa,GrupodeInvestigaci´onFQM0229.12M.´ALVAREZDEMORALES,L.FERN´ANDEZ,T.E.P´EREZ,ANDM.A.PI˜NARThespecialshapeofthepolynomialsinvolvedintheaboveequationisadirectconsequenceofthefactthateveryorthogonalpolynomialoftotaldegreenmustsatisfythesamePDE.KrallandSheffershowedthat,uptoalinearchangeinthevariables,thereareninedifferentsetsoforthogonalpolynomialssatisfyingsuchtypeofPDE.ThefirstreferencetoaRodriguesformulaforclassicalorthogonalpolynomialsintwovariablesappearsintheclassicaltextbyP.AppellandJ.Kamp´edeF´eriet([1]).Later,P.K.Suetin([16]),andY.J.Kim,K.H.KwonandJ.K.Lee([10])considerananalogueoftheRodriguesformulaforKrallandShefferclassicalorthogonalpolynomialsintwovariables.Infact,fornapositiveinteger,theydefine(3)Pn−i,i(x,y)=1ω∂n−ix∂iy(pn−iqiω),wherew(x,y)isaweightfunctionoverasimplyconnecteddomain,andasymmetryfactorofL,thelineardifferentialoperatordefinedin(2),andp(x,y),q(x,y)arepolynomialsrelatedwiththepolynomialcoefficientsin(2).Then,undersomead-ditionalhypothesis,(3)definesanalgebraicpolynomialintwovariablesorthogonaltoallpolynomialsoflowerdegree(seeexample3,inSection6).TheaboveRodriguesformularunsonlyforclassicalorthogonalpolynomialsas-sociatedwithapositivedefinitemomentfunctional,sinceitneedsaweightfunction.Nevertheless,H.L.KrallandI.M.Shefferfoundedclassicalorthogonalpolynomialsintwovariablesassociatedwithanonpositivedefinitemomentfunctionalwhichhasasymmetryfactor(seeL.L.Littlejohn[15]),butnotaRodriguesformulalike(3)([10]).Ontheotherhand,tensorproductoftwoclassicalorthogonalpolynomialsinonevariable,definedbyPh,k(x,y)=Rh(x)Sk(y),h,k≥0,where{Rh}h≥0and{Sk}k≥0areHermite,Laguerre,JacobiorBesselpolynomials,satisfiesaRodriguesformulaas(3).Infact,Ph,k(x,y)canbewrittenasaproductoftherespectiveRodriguesformulasPh,k(x,y)=1ω1∂hx(φh1ω1)1ω2∂ky(φk2ω2),h,k≥0.However,tensorproductsofclassicalorthogonalpolynomialsinonevariablearenotclassicalaccordingtotheKrallandShefferdefinitionsincetheydonotsatisfyequation(2),exceptforHermiteandLaguerrepolynomials.Recently,theauthors(see[5,6,7,8])extendedtheconceptofclassicalorthogonalpolynomialsintwovariablestoawiderframework,which,ofcourse,includestheKrallandShefferdefinitionandtensorproductsofclassicalorthogonalpolynomialsinonevariable.Thevectorrepresentationfororthogonalpolynomialsintroducedin[12,13],anddevelopedin[17]isthekeytointroducetheconceptofclassicalorthogonalpoly-nomialsintwovariables.Let{Pn}ndenoteaweakorthogonalpolynomialsequence(seeSection2),itwillbecalledclassical(inanextendedsense)ifthereexistnonsingularmatricesΛn∈Mn+1(R),suchthat,(4)L[Ptn]≡div(Φ∇Ptn)+˜Ψt∇Ptn=PtnΛn,ARODRIGUESFORMULAINTWOVARIABLES3whereΦ=abbc,˜Ψ=d−ax−bye−bx−cy,anda,b,carepolynomialsintwovariablesoftotaldegreelessthanorequalto2,andd,earepolynomialsintwovariablesoftotaldegreelessthanorequalto1,anddivand∇denotetheusualdivergenceandgradientoperatorsintwovariables.Observethatthelefthandsideof(4)generalizesthelefthandsideoftheKrallandShefferPDE(2),withoutanyrestrict