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1、RationalsolutionsofthediscretetimeTodalatticeandthealternatediscretePainlev´eIIequationAlanK.CommonandAndrewN.W.Hone††InstituteofMathematics,Statistics&ActuarialScience,UniversityofKent,CanterburyCT27NF,UKE-mail:A.N.W.Hone@kent.ac.ukAbstract.TheYablonskii-Vorob’evpolynomialsyn(t),whicharedefinedby
2、asecondorderbilineardifferential-differenceequation,providerationalsolutionsoftheTodalattice.Theyarealsopolynomialtau-functionsfortherationalsolutionsofthesecondPainlev´eequation(PII).Herewedefinetwo-variablepolynomialsYn(t,h)onalatticewithspacingh,byconsideringrationalsolutionsofthediscretetimeToda
3、latticeasintroducedbySuris.ThesepolynomialsareshowntohavemanypropertiesthatareanalogoustothoseoftheYablonskii-Vorob’evpolynomials,towhichtheyreducewhenh=0.TheyalsoproviderationalsolutionsforaparticulardiscretisationofPII,namelythesocalledalternatediscretePII,andthisconnectionleadstoanexpressionin
4、termsoftheUmemurapolynomialsforthethirdPainlev´eequation(PIII).ItisshownthatB¨acklundtransformationforthealternatediscretePainlev´eequationisasymplecticmap,andtheshiftintimeisalsosymplectic.FinallywepresentaLaxpairforthealternatediscretePII,whichrecoversJimboandMiwa’sLaxpairforPIIinthecontinuumli
5、mith→0.Submittedto:J.Phys.A:Math.Theor.arXiv:0807.3731v4[nlin.SI]23Sep2008RationalsolutionsofdiscreteTodaandalt-dPII21.IntroductionTheTodalatticed2xn=exn−1−xn−exn−xn+1,n∈Z(1)dt2wasthefirstintegrabledifferential-differenceequationtobediscovered[1].TheYablonskii-Vorob’evpolynomials[2,3]yieldrationalso
6、lutionsofboththeTodalatticeandthesecondPainlev´etranscendent(PII),sincethetau-functionsofPIIsatisfythebilinearformoftheTodalattice.Inapreviouswork[4]oneoftheauthorsobtainedanexpressionforsolutionsoftheTodalatticeasratiosofHankeldeterminants,byusingtheassociatedLaxpairtoconstructcontinuedfractions
7、olutionstoasequenceofRiccatiequations.ThisinturnledtoanexpressionfortheYablonskii-Vorob’evpoynomialsasHankeldeterminants[5],equivalenttothatdiscoveredmorerecently[6](seealso[7]).Herewewillconsiderthecasewhenthetimeevol
