Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

Lance L. Littlejohn; Samuel D. Shore

doi:10.4153/CMB-1982-040-2

Abstract

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One of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:

In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

References

1. Cole, R. H., Theory of Ordinary Differential Equations, Appleton-Century-Crofts, New York, 1968.Google Scholar

2. Hahn, W., On Differential Equations for Orthogonal Polynomials, Funkcialaj Ekvacioj, Vol. 21 (1978), 1-9.Google Scholar

3. Krall, A. M., Orthogonal Polynomials Satisfying Fourth Order Differential Equations, Proc. Royal Soc. Edin. (to appear).Google Scholar

4. Krall, A. M. and Morton, R. D., Distributional Weight Functions for Orthogonal Polyonmials, S.I.A.M. J. Math. Anal., 4 (1978), 604-626:Google Scholar

5. Krall, H. L., On Orthogonal Polynomials satisfying a certain Fourth Order Differential Equation, The Pennsylvania State College Studies, No. 6, The Pennsylvania State College, State College, PA, 1940.Google Scholar

6. Shore, S. D., On the Sets of Orthogonal Polynomials which satisfy a Second Order Differential Equation, The Pennsylvania State University, Master's Thesis, 1961.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Littlejohn, Lance L. 1983. The Krall Polynomials as Solutions to a Second Order Differential Equation. Canadian Mathematical Bulletin, Vol. 26, Issue. 4, p. 410.

Magnus, Alphonse 1984. Padé Approximation and its Applications Bad Honnef 1983. Vol. 1071, Issue. , p. 213.

Littlejohn, Lance L. 1984. Differential Equations, Proceedings of the Conference held at The University of Alabama in Birmingham. Vol. 92, Issue. , p. 413.

Littlejohn, Lance L. 1984. On the classification of differential equations having orthogonal polynomial solutions. Annali di Matematica Pura ed Applicata, Vol. 138, Issue. 1, p. 35.

Koornwinder, Tom H. 1984. Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1). Canadian Mathematical Bulletin, Vol. 27, Issue. 2, p. 205.

Hahn, Wolfgang 1985. Polynômes Orthogonaux et Applications. Vol. 1171, Issue. , p. 16.

Hendriksen, E and van Rossum, H 1985. A Padé-type approach to non-classical orthogonal polynomials. Journal of Mathematical Analysis and Applications, Vol. 106, Issue. 1, p. 237.

Littlejohn, Lance L. 1986. AN APPLICATION OF A NEW THEOREM ON ORTHOGONAL POLYNOMIALS AND DIFFERENTIAL EQUATIONS. Quaestiones Mathematicae, Vol. 10, Issue. 1, p. 49.

Littlejohn, Lance L. and Krall, Allan M. 1989. Orthogonal polynomials and higher order singular Sturm-Liouville systems. Acta Applicandae Mathematicae, Vol. 17, Issue. 2, p. 99.

Marcellan, F. and Maroni, P. 1992. Sur l'adjonction d'une masse de Dirac á une forme régulière et semi-classique. Annali di Matematica Pura ed Applicata, Vol. 162, Issue. 1, p. 1.

Pasquini, Lionello 1994. Approximation and Computation: A Festschrift in Honor of Walter Gautschi. p. 511.

Koekoek, Roelof 1994. Differential equations for symmetric generalized ultraspherical polynomials. Transactions of the American Mathematical Society, Vol. 345, Issue. 1, p. 47.

Campos, R. G. and Avila, L. A. 1995. Some properties of orthogonal polynomials satisfying fourth order differential equations. Glasgow Mathematical Journal, Vol. 37, Issue. 1, p. 105.

Grünbaum, F.Alberto 1998. Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials. Journal of Computational and Applied Mathematics, Vol. 99, Issue. 1-2, p. 189.

Kwon, K.H. Littlejohn, L.L. and Yoon, G.J. 2006. Construction of differential operators having Bochner–Krall orthogonal polynomials as eigenfunctions. Journal of Mathematical Analysis and Applications, Vol. 324, Issue. 1, p. 285.

Cox, Ben Futorny, Vyacheslav and Tirao, Juan A. 2013. DJKM algebras and non-classical orthogonal polynomials. Journal of Differential Equations, Vol. 255, Issue. 9, p. 2846.

Bryenton, Kyle R. Cameron, Andrew R. Kirk, Keegan L. A. Saad, Nasser Strongman, Patrick and Volodin, Nikita 2020. On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application. Algorithms, Vol. 13, Issue. 11, p. 286.

Magnus, Alphonse P. Ndayiragije, François and Ronveaux, André 2021. About families of orthogonal polynomials satisfying Heun’s differential equation. Journal of Approximation Theory, Vol. 263, Issue. , p. 105522.

Article contents

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

Abstract

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References

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

Abstract

Keywords

References

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