Transformation formulae for multivariable basic hy

arXiv:math/9803146v1[math.QA]30Mar1998TransformationformulaeformultivariablebasichypergeometricseriesT.H.BakerandP.J.ForresterDepartmentofMathematics,UniversityofMelbourne,Parkville,Victoria3052,AustraliaWestudymultivariable(bilateral)basichypergeometricseriesassociatedwith(typeA)Macdonaldpolynomials.Wederiveseveraltransformationandsummationprop-ertiesforsuchseriesincludinganaloguesofHeine’s2φ1transformation,theq-Pfaff-KummerandEulertransformations,theq-Saalsch¨utzsummationformulaandSear’stransformationforterminating,balanced4φ3series.Forbilateralseries,werederiveKaneko’sanalogueofthe1ψ1summationformulaandgivemultivariableextensionsofBailey’s2ψ2transformations.DedicatedtoDickAskeyontheoccasionofhis65’thbirthday.0IntroductionMultivariablebasichypergeometricseriesofthetypestudiedinthispaperwerefirstintroducedbyKanekoandMacdonald[14,17].TheyaredefinedasrΦsa1,...,arb1,...,bs;z:=Xλ(−1)|λ|qn(λ′)s+1−r(a1)λ···(ar)λ(b1)λ···(bs)λh′λPλ(z;q,t)(0.1)whereallquantitiesaredefinedinSection1.Theq=1casehadbeenstudiedpreviouslybyYan[22]whereseveralimportantpropertiesofthehypergeometricseriespFqwerestudiedincludingtheGauss(2F1)andKummer(1F1)summationformulae,thePfaff-KummerandEulertransformationsandintegralrepresentationswerederived(seealso[20]fortheGaussformulawithrespecttoarbitraryrootsystems).Kaneko[14]consideredageneralizedq-Selbergintegraldependentonparametersx1,...,xmandderivedasetofmq-differenceequationssatisfiedbysuchanintegral.Hethenshowedthatthebasichypergeometricseries2Φ1definedby(0.1)wastheuniquesolution(satisfyingcertainproperties)ofsuchasystem.Hewasthusabletoderiveanintegralrepresentationforthis2Φ1seriesandhencegiveanalternativeproofoftheq-Selbergintegral[2](seealso[9,23,10]).Headditionallyderivedtheq-analogueoftheGaussformulaandanotherintegralformula,theconstanttermversionofwhichwaspresentedin[16,Theorem4].Inasubsequentwork[12],KanekointroducedamultivariableanalogueofthebilateralbasichypergeometricseriesrΦsandderivedananalogueofRamanujan’s1ψ1summationformula,alongwithamultivariableversionoftheJacobitripleproductidentity(whichisalimitingcase-seealso[15]).Independently,Macdonaldinhisunpublishednotes[17]carriedout(amongotherthings)asimilarprogramtoYanintheq=1casealongwithamultivariableversionoftheSaalsch¨utzsummationformula(thesummationofabalanced3F2withunitargument),whileforgeneralqhederivedtheintegralrepresentationforthe2Φ1series,andq-analoguesoftheGaussandSaalsch¨utzformulae.Theaimofthepresentworkistosupplementsomeoftheexistingknowledgewithsomenewtransformationandsummationformulaeinthismultivariablesetting(includingtheq-analogueofthePfaff-Kummertransformationformulae,Heine’stransformationformulaforthe12Φ1series,Sear’stransformationforterminating,balanced4Φ3series,andvarioussummationandtransformationformulaeforbilateral2Ψ2series),aswellasprovidingalternativederivationsofknownresults.Foracomprehensivereviewofsummationandtransformationformulaeforbasichypergeometricseriesintheone-variablecaseseeGasperandRahmen’sbook[7].Wenoteherethatmanyofthesummationandtransformationformulaeinvolvingmulti-variablebasichypergeometricseriesareonlyvalidwhentheargumentisspecializedtoztδ:=(z,zt,zt2,...,ztn−1)(theexceptionsbeingtheq-binomialtheorem,theEulertransformationofthe2Φ1seriesandthe1Ψ1summationformula).Finally,wepointoutthatmanyoftheformulaepresentedherecanbederivedinthecaseq=t(theSchurpolynomialcase)asspecialcasesoftheverygeneralformulaeofMilneandco-workers(see[19]andreferencedtherein),orfromthosefoundintheworksofKrattenthaler,GustafsonandSchlosser[8,21].Also,summationformulaeforhypergeometricsystemsassociatedwiththeBCrootsystemhavebeenconsideredbyVanDiejen[4](seealso[5]).Theplanofthepaperisasfollows.InSection1wesetoutthebasicfactsaboutMacdon-aldpolynomialsweshallrequire.Section2exhibitsamultivariableextensionofHeine’s2φ1transformations.TheEulertransformationisderivedinSection3fromthedefiningdifferenceequationsforthe2Φ1series.Section4treatstheq-Saalsch¨utzformulawhiletheq-Pfaff-Kummertransformationfor2Φ1andSear’stransformationforterminating,balanced4Φ3seriesaredis-cussedinSection5.BilateralseriesarestudiedinSection6wherethe1Ψ1summationformulaisderivedusingashiftedversionofthemultivariableGausssummationformula,andvarious2Ψ2transformationsarederivedusingtwodifferentmethods.1NotationsTheMacdonaldpolynomialsPλ(x;q,t),x:=(x1,...,xn)(oftenabbreviatedtoPλ(x)orjustPλwhenthecontextisclear)aredefinedastheuniquesymmetricpolynomialshavingtheexpansionPλ(x)=mλ(x)+Xμλcλμmμ(x)(wheredenotesthedominanceorderonpartitionsandmλ(x)denotesthemonomialsymmetricfunction),whichformanorthogonalbasisofsymmetricfunctionswithrespecttotheinnerproducthf,gi:=1n!C.T.f(x)g(x−1)Δq(x)
(1.1)whereΔq(x):=Y1≤ij≤n(xi/xj;q)k(xj/xi;q)k(1.2)HereC.T.standfor“theconstanttermof”inthecasewheret=qk(andhencethequantityinbracketsin(1.1)isaLaurentpolynomial)orthecorrespondingtrigonometricintegralotherwise,and(x;q)a:=(x)∞(xqa;q)∞,(u;q)∞:=∞Yi=0(1−uqi).Definethefollowingquantitieshλ=Ys∈λ(1−qa(s)tl(s)+1)h′λ=Ys∈λ(1−qa(s)+1tl(s))(1.3)wherea(s)(respectivelyl(s))denotesthearm-length(resp.leg-lengt