法推导二阶差分方程式渐进解

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1、ASteepestDescentMethodfortheAsymptoticSolutions1ofaSecond-OrderDifferenceEquation最陡下降法推導二階差分方程式漸進解ASteepestDescentMethodfortheAsymptoticSolutionsofaSecond-OrderDifferenceEquation周章JangJou國際企業管理系吳鳳技術學院DepartmentofInternationalBusinessManagementWuFengInstituteofTechnology摘要：就一個由二廣義拉格爾

2、函數及一些基本函數之乘積所構成的積分，考慮其所滿足之二階差分方程式。此方程式之一對線性獨立解乃係此差分方程式之最小解及主勢解，且與合流超幾何函數Φ及Ψ有關。據此，本文利用最陡下降法推導出此方程式之一對線性獨立解分別的漸進公式。關鍵詞：最陡下降法，漸進表示式，合流超幾何函數Abstract:Usingthemethodofsteepestdescent,asymptoticformulasareobtainedforapairoflinearlyindependentsolutionsofasecond-orderdifferenceequationwhichi

3、ssatisfiedbyanintegralinvolvingtheproductoftwogeneralizedLaguerrepolynomialswithsomeelementaryfunctions.ThepaircorrespondtominimalanddominantsolutionsofthedifferenceequationandarerelatedtotheconfluenthypergoemetricfunctionsΦandΨ.Keywords:MethodofSteepestDescent,AsymptoticRepresentat

4、ion,ConfluentHypergeometricFunction.1.IntroductionConsiderthesecond-orderdifferenceequationy+ay+by=0(n=1,2,3,L)(1-1)n+1nnnn−1whereanandbnareasetofgivenrealnumberswithbn≠0.Thegeneralsolutionof(1-1)canberepresentedasalinearcombinationofapairoflinearlyindependentsolutionsfandg.Itisknow

5、n[5]thatifthetwosolutionsaresuchthatnnfnlim=0(1-2)n→∞gnthenseriousproblemofnumericalinstabilitywilloccurwhenonetriestocomputethe~337~2吳鳳學報第15期solutionf,oranyconstantmultipleofitbyusingtherecurrencerelation(1-1)inforwardndirection.Asimpleandeffectivewaytoremedythisproblemistoapplythe

6、recurrencerelationinbackwarddirection.ThisisfirstintroducedforthecomputationofmodifiedBesselfunctionsIn(x)andisknownastheMilleralgorithm[3].Solutionssuchasfnareoftencalledminimalsolutions[6],whilesolutionssuchasgnwhichsatisfy(1-2)withrespecttoaminimalsolutionfarecalleddominant.Forar

7、eviewonthistechnique,seenforexample[7,9,11,12,19,20,22,23,25].WhenimplementingtheMilleralgorithmtocomputeaminimalsolution,oneusuallychoosestostartwiththeinitialvaluesy=0and0y=forasufficientlylargenn−1n.Theseinitialvaluesdonotcorrespondtothetruevaluesofthesolution.Questionsthereforen

8、aturallyariseastowh