Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential Equations

Abstract

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. J., 4 (1961/1962), 318–331 24:B2554 0103.34203 CrossrefISIGoogle Scholar[5] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205, England 37:3733 Google Scholar[6] K. O. Geddes, Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE'S, Proc. 1977 MACSYMA Users' Conference, Univ. of California, Berkeley, CA, 1977, 405–423, NASA CP-2012 Google Scholar[7] T. S. Horner, Recurrence relations for the coefficients in Chebyshev series solutions of ordinary differential equations, Math. Comp., 35 (1980), 893–905 81d:65038 0446.65040 CrossrefISIGoogle Scholar[8] S. Lewanowicz, Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series, Zastos. Mat., 15 (1976), 345–396 54:6527 0357.33006 Google Scholar[9] S. Lewanowicz, Construction of the lowest-order recurrence relation for the Jacobi coefficients, Zastos. Mat., 17 (1983), 655–675 85d:33030 0591.65089 Google Scholar[10] Stanisław Lewanowicz, Recurrence relations for hypergeometric functions of unit argument, Math. Comp., 45 (1985), 521–535 86m:33004 0583.33005 CrossrefISIGoogle Scholar[11] Y. L. Luke, The Special Functions and their Approximations, Academic Press, New York, 1969 Google Scholar[12] A. Magnus, Application des récurrences au calcul d'une classe d'intégrales, Rep., 71, Inst. Math. Pure Appl., Univ. de Louvain, 1974 Google Scholar[13] A. G. Morris and , T. S. Horner, Chebyshev polynomials in the numerical solution of differential equations, Math. Comp., 31 (1977), 881–891 56:1729 0386.65040 CrossrefISIGoogle Scholar[14] O. Oluremi Olaofe, On the Tchebyschev method of solution of ordinary differential equations, J. Math. Anal. Appl., 60 (1977), 1–7 10.1016/0022-247X(77)90043-9 56:1724 0363.65065 CrossrefISIGoogle Scholar[15] Stefan Paszkowski, Zastosowania numeryczne wielomianów i szeregów Czebyszewa, Państwowe Wydawnictwo Naukowe, Warsaw, 1975, 481– 56:13534 0423.65012 Google Scholar[16] N. Robertson, An ALTRAN program for finding a recursion formula for the Gegenbauer coefficients of a function, Spec. Rep., SWISK 11, Nat. Res. Inst. for Math. Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics