Macdonald Polynomials and Multivariable Basic Hype

arXiv:math/0611639v2[math.CO]30Mar2007Symmetry,IntegrabilityandGeometry:MethodsandApplicationsSIGMA3(2007),056,30pagesMacdonaldPolynomialsandMultivariableBasicHypergeometricSeries⋆MichaelJ.SCHLOSSERFakult¨atf¨urMathematik,Universit¨atWien,Nordbergstraße15,A-1090Vienna,AustriaE-mail:[e-mail]:∼schlosse/ReceivedNovember21,2006;PublishedonlineMarch30,2007Originalarticleisavailableatφ5summationformula.WederiveseveralnewrelatedidentitiesincludingmultivariateextensionsofJackson’svery-well-poised8φ7summation.Motivatedbyourbasichypergeometricanalysis,weproposeanextensionofMacdonaldpolynomialstoMacdonaldsymmetricfunctionsindexedbyparti-tionswithcomplexparts.Theseappeartopossessniceproperties.Keywords:Macdonaldpolynomials;Pieriformula;recursionformula;matrixinversion;basichypergeometricseries;6φ5summation;Jackson’s8φ7summation;An−1series2000MathematicsSubjectClassification:33D52;15A09;33D671IntroductionTheobjectiveofthispaperistostudysomeaspectsofAn−1Macdonaldpolynomials(whichareafamilyofsymmetricmultivariableorthogonalpolynomialsassociatedwiththeirreduciblereducedrootsystemAn−1,introducedbyI.G.Macdonald[32]inthe1980’s),withaparticularemphasisontheirconnectionto(multivariable)basichypergeometricseries.MacdonaldpolynomialsoftypeAareindexedbyintegerpartitions,andformabasisofthealgebraofsymmetricfunctionswithrationalcoefficientsintwoparametersqandt.Theygeneralizemanyclassicalbasesofthisalgebra,includingmonomial,elementary,Schur,Hall–Littlewood,andJacksymmetricfunctions.Theseparticularcasescorrespondtovariousspe-cializationsoftheindeterminatesqandt.Intermsofbasichypergeometricseries,theMacdon-aldpolynomialscorrespondtoamultivariablegeneralizationofthecontinuousq-ultrasphericalpolynomials,see[25].Aprincipaltoolforstudyingq-orthogonalpolynomials(seee.g.[18])isthetheoryofbasichypergeometricseries(cf.[12]),richofidentities,havingapplicationsindifferentareassuchascombinatorics,numbertheory,statistics,andphysics(cf.[1]).Hypergeometricandbasichypergeometricseriesundoubtedlyplayaprominentroleinspecialfunctions,see[2].Eveninonevariable,theyarestillanobjectofactiveresearch.Anotablerecentadvanceincludeselliptic(ormodular)hypergeometricseries(surveyedin[12,Ch.11]and[52])whichisaone-parametergeneralizationofbasichypergeometricseries,firstintroducedbyFrenkelandTuraev[11]inastudyrelatedtostatisticalmechanics.Aconvenienttool(suggestedhere)forfurtherdevelopingthetheoryofmultivariableq-orthogonalpolynomialsisthetheoryofmultivariablebasichypergeometricseriesassociatedwith⋆ThispaperisacontributiontotheVadimKuznetsovMemorialIssue‘IntegrableSystemsandRelatedTopics’.Thefullcollectionisavailableat(or,equivalently,withLiealgebras).BasichypergeometricseriesassociatedwiththerootsystemAn−1(orequivalently,associatedwiththeunitarygroupU(n))havetheiroriginintheworkofthethreemathematicalphysicistsHolman,Biedenharn,andLouck,startingin1976,see[16,17].Theirworkwasdoneinthecontextofthequantumtheoryofangularmomentum,usingmethodsrelyingontherepresentationtheoryoftheunitarygroupU(n).Subsequently,extensiveinvestigationsinthetheoryofmultiplebasichypergeometricseriesassociatedtotherootsystemAn−1havebeencarriedoutbyR.A.Gustafson,S.C.Milne,andlatervariousotherresearchers.Asresult,manyoftheclassicalformulaeforbasichypergeometricseriesfrom[12]havealreadybeengeneralizedtothesettingofAn−1series(seeSubsection3.2forsomeselectedresults).AnimportantresultthatconnectsAn−1basichypergeometricserieswithMacdonaldpoly-nomialsisKajiharaandNoumi’s[22]explicitconstructionofraisingoperatorsofrowtypeforMacdonaldpolynomials.TheirconstructionutilizedAn−1terminating1φ0and2φ1summationspreviouslyobtainedbyMilne[40](which,however,werederivedindependentlyin[22]usingMacdonald’sq-differenceoperator).InthispaperwerevealyetmoreconnectionsofAn−1basichypergeometricserieswithMac-donaldpolynomials.Ontheotherhand,wealsounderstandthepresentworkasacontributiontowardsthedevelopmentofatheoryofmultivariablevery-well-poisedbasichypergeometricseriesinvolvingMacdonaldpolynomialsoftypeA.VariousidentitiesformultiplebasichypergeometricseriesofMacdonaldpolynomialargumenthavebeenderivedbyMacdonald[33,p.374,Eq.(4)],Kaneko[23,24],BakerandForrester[3],andWarnaar[54].Theseauthorsinfactderivedmultivariableanaloguesofmanyoftheclassicalsummationandtransformationformulaeforbasichypergeometricseries.Asamatteroffact,noneofthesemultivariateidentitiesreducetosummationsortransformationsforvery-well-poisedbasichypergeometricseriesintheunivariatecase.Therearethusseveralotherclassicalbasichypergeometricidentitiesforwhichhigher-dimensionalextensionsinvolvingMacdonaldpolynomialsoftypeAhave