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1、ARepresentationoftheSolutionofthenthOrderLinearDifferenceEquationWithVariableCoefficients*R.KitKittappaMillersvilleUniversityMillersville,Pennsylvania17551SubmittedhvHansSchneiderABSTRACTThelineardifferenceequationofthenthorderwithvariablecoefficientsandarelateddifferenceequationareconsidered.Itiss
2、hownthat,foreachproblem,thedeterminantsofsubmatricesofasinglesolutionmatrixprovidesthesolutionse-quence.Applicationstosequencesandseriesareshown.ResultsonToeplitzmatricesinvolvingtherootsofassociatedpolynomialequationsareobtained.1.INTRODUCTIONDifferenceequationsareusuallysolvedbymethodsanalgoustot
3、hoseappliedinsolvmgdifferentialequations111,becausetheyarethediscreteanalogsofdifferentialequations.Thelargenumberofapparentlyunrelatedmethodsfoundintextbooksforsolvingdifferenttypesofordinarydifferentialequationsaretranscribedintomethodstosolvedifferenceequations.(Liegroupmethodsbasedonusingsymmet
4、riesofdifferentialequations[4,51canbringoutrelationsbetweenthemethodsusedtosolvedifferentialequations,butitdoesntoappearthatsuchdiscussionshavebeenextended,atleastnotextensively,todifferenceequations.1Weuseauniformapproach,basedonlinearalgebra,tosolvelinearnthorderdifferenceequationswithvariablecoe
5、fficients.*PartlybasedonapaperpresentedattheNSF/CBMSConferenceonStructuredMatrixTheoryatAtlanta,Ga.(Aug.1991).LINEARALGEBRAANDITSAPPLICATIONS193:211-222(1993)2110ElsevierSciencePublishingCo.,Inc.,1993655AvenueoftheAmericas,NewYork,NY100100024-3795/93/$6.00212R.KITKITTAPPAWeconsiderthedifferenceequa
6、tioniP(k3i)Yi=f(k),(1)i=lwherethep’sandfareknownfunctionswithp(k,k)z0,andkcantakeonthevalues1,2,3,...;wealsoconsidertheequation2dk,i+k1yi.k=g(k),i=oalongwiththeinitialconditionsYj=cjaj=1,2,3,-..,n,(2b)wheretheq’sandgareknownfunctionswithq(k,n+k)#0,andkcantakeonthevalues,1,2,3,....Equations(2a)and(2
7、b)representaninitialvalueprobleminvolvingannthorderlineardifferenceequationwithvariablecoefficients,andequation(1)isacommonlyoccurringdifferenceequation.Thesolutionforeachproblemisasequenceyk(k=1,2,3,...>.W
