Differential properties of matrix orthogonal polyn

arXiv:math/0205094v1[math.CA]9May2002DIFFERENTIALPROPERTIESOFMATRIXORTHOGONALPOLYNOMIALSM.J.Cantero,L.Moral,L.Vel´azquezDepartamentodeMatem´aticaAplicada,UniversidaddeZaragoza,SpainDecember2001AbstractInthispaperageneraltheoryofsemi-classicalmatrixorthogonalpolynomialsisdeveloped.Wedefinethesemi-classicallinearfunctionalsbymeansofadistri-butionalequationD(uA)=uB,whereAandBarematrixpolynomials.Severalcharacterizationsforthesesemi-classicalfunctionalsaregivenintermsofthecor-responding(left)matrixorthogonalpolynomialssequence.Theyinvolveaquasi-orthogonalitypropertyfortheirderivatives,astructurerelationandasecondorderdiffero-differentialequation.Finallyweillustratetheprecedingresultswithsomenon-trivialexamples.Keywordsandphrases:Orthogonalmatrixpolynomials,Semi-classicalfunctionals,Differentialequation,Structurerelation.(2000)AMSMathematicsSubjectClassification:42C05Suggestrunninghead:DifferentialpropertiesofmatrixOP.Correspondingauthor:[e-mail]:[phone]Fax:[phone]ThisworkwassupportedbyDirecci´onGeneraldeEnse˜nanzaSuperior(DGES)ofSpainundergrantPB98-1615.TypesetbyAMS-TEX12DIFFERENTIALPROPERTIESOFMATRIXORTHOGONALPOLYNOMIALS§1-Introduction.Thestudyofsemi-classicalorthogonalpolynomialsinthescalarcase(i.e.,thestudyoforthogonalpolynomialwhoseassociatedfunctionalsatisfiesadistribu-tionalequationD(uΦ)=uΨ,whereΦ,ΨarepolynomialswithdegΨ≥1)wasstartedbyShohat([15])inordertogeneralizethepropertiesofclassicalorthogonalpolynomials.Amongothers,in([2,9]),anapproachtosuchpolynomialstakingintoaccountthequasi-orthogonalityofthederivativesofthepolynomialssequenceisgiven.WealsomentiontheresultsofMaroni([11,12])whereanalgebraictheoryofsemi-classicalorthogonalpolynomialsispresented.Thepurposeofthistheoryisthecharacterizationofsemi-classicalorthogonalpolynomialsbymeansofthequasi-orthogonalityoftheirderivatives,thestructurerelationandthediffero-differentialequationofsecondorder.Inthelastyears,thestudyofmatrixorthogonalpolynomialsattractedagreatinterestoftheresearchers,(see[4,6,10]).AsfortheirdifferentialpropertiesitisknowntheresultofDur´an([5])whocharacterizesthosematrixorthogonalpolyno-mials(OP)satisfyingasymmetricsecondorderdifferentialequationwithpolyno-mialcoefficients.Heprovesthattheyarediagonal(uptoafactor)withclassicalscalarOPinthediagonal.Inspiteofthevarietyofapplicationsofmatrixpoly-nomials([4,5,6]),therearenottoomanyknownfamiliesofsemi-classicalmatrixOPoutofthediagonalcase.OnewaytostudymanyfamiliesofmatrixorthogonalpolynomialsistoextendtheanalysisstartedbyDur´anofdifferentialpropertiesofmatrixorthogonalpoly-nomials.Anaturalwaytodothisistogeneralizethetheoryofsemi-classicalscalarOPtothematricialcase.Inordertodothat,westartwithaPearsontypeequa-tionformatrixquasi-definitefunctionals.Weobtaintheircharacterizationintermsofa(ingeneral,non-symmetric)differo-differentialequationfortherelatedmatrixorthogonalpolynomials.Inthewayoftheproof,wefindanotherequivalencesthatgeneralizethescalarcase.Thispaperhasbeenorganizedasfollows.InSection3weintroduceandstudytheconceptofquasi-orthogonalityformatrixpolynomials.ThemeanresultofSection4isthecaracterizationofsemi-classicalfunctionalsintermsofthequasi-orthogonalityofthecorrespondingmatrixpolynomialsandtheirderivatives.InSection5weproveformatrixOPwiththeabovequasi-orthogonalityproper-ties,thestructurerelationandthediffero-differentialequation,showingthattheyareequivalenttoaPearson-typeequation.Finally,inSection6,weillustratetheprecedingwithsomeexamplesandwefindawaytoconstructnon-diagonalizablesemi-classicalmatrixfunctionals.§2-Basictools.WeshalldenotebyP(m)theC(m,m)-left-moduleP(m)=(nXk=0αkxkαk∈C(m,m);n∈N)M.J.CANTERO,L.MORAL,L.VEL´AZQUEZ3andbymeansofP(m)′theC(m,m)-right-moduleHomP(m),C(m,m)
.2.1Definition.(i)Thedualitybracketisdefinedby·,·:P(m)×P(m)′→C(m,m)P,u
→P,u:=uP
(ii)Fork∈Nandu∈P(m)′thelinearfunctionaluxkI∈P(m)′isdefinedbyP,uxkI:=xkP,u,whereIdenotesthem×midentitymatrix.Alinearextensiongivestheright-productu·A∈P(m)′foru∈P(m)′,A∈P(m),withA(x)=nXk=0αkxk,inthefollowingway:P,uA=nXk=0xkP,uαk.(iii)Theinnerproduct·,·:P(m)×P(m)→C(m,m)P,Q
→P,Qu:=P,uQ∗isdefinedforeveryu∈P(m)′,whereQ∗denotesthetrasposedconjugatedofQ.Remark.Thedualitybracketverifiesthefollowinglinearproperties:α1P1+α2P2,β1u1+β2u2==α1P1,u1β1+α1P1,u2β2+α2P2,u1β1+α2P2,u2β2,forallα1,α2,β1,β2∈C(m,m),P1,P2∈P(m)andu1,u2∈P(m)′,whiletheinnerproductissesquilinear:(i)α1P1+α2P2,β1Q1+β2Q2u=α1P1,Q1uβ∗1++α1P1,Q2uβ∗2++α2P2,Q1uβ∗1+α2P2,Q2uβ∗2,forallα1,α2,β1,β2∈C(m,m),P1,P2,Q1,Q2∈P(m);(ii)P,Q∗u=Q,PuforallP,Q∈P(m).2.2Definition.WedenotebyCk:=xkI,uthek-thmomentwithrespecttou∈P(m)′.Givenu∈P(m)′withmoments(Ck)k∈N,wesaythatuisquasi-definite(non-singular)ifdetCk+j
nk,j=06=0,∀n≥0,were(Ck+j)nk,j=0istheHankel-blockmatrixC0C1···CnC1C2···Cn+1············CnCn+1···C2n4DIFFERENTIALPROPERTIESOFMATRIXORTHOGONALPOLYNOMIALSRemark.GiventhesequenceCk
∞k=0⊂C(m,m),thereexistsauniqueu∈P(m)′suchthatxkI,u=Ck.ThisestablishesanisomorphismbetweenP(m)′andtheformalserieswithcoefficientsinC(m,m),∞Xk=0Ckxk.2.3Definition.Letu∈P(m)′.Wesaythatuishermitiani