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Some properties of orthogonal polynomials satisfying fourth order differential equations

  • R. G. Campos (a1) and L. A. Avila (a1)
Abstract

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

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References
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1.Campos R. G., Some Properties of the Zeros of Polynomial Solutions of Sturm-Liouville Equations, SI AM J. Math. Anal. 18 (1987), 16641668.
2.Dehesa J. S., Buendía E. and Sánchez-Buendía M. A., On the polynomial solutions of ordinary differential equations of the fourth order, J. Math. Phys. 26 (1985), 15471552.
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4.Krall A. M., A review of orthogonal polynomials satisfying boundary value problems, in “Orthogonal Polynomial and Their Applications”, (Alfaro M. et al. , eds.), Lect. Notes in Math., 1329, Springer-Verlag, Berlin (1988), 7397.
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6.Littlejohn L. L., Orthogonal polynomial solutions to ordinary and partial differential equations, In “Orthogonal Polynomial and Their Applications”, (Alfaro M. et al. , eds.), Lect. Notes in Math., 1329, Springer-Verlag, Berlin (1988), 98124.
7.Littlejohn L. L. and Krall A. M., Orthogonal polynomials and higher order singular Sturm-Liouville systems, Acta Appl. Math. 17 (1989), 99170.
8.Littlejohn L. L. and Shore A. M., Nonclassical orthogonal polynomials as solutions to second order differential equations, Canad. Math. Bull. 25 (1982), 291295.
9.Popoviciu T., Sur certains problémes de maximum de Stieltjes, Bull. Math. Soc. Roumaine Sci., 38 (1936), 7396.
10.Szegö G., Orthogonal polynomials (American Mathematical Society, Providence, RI, fourth edition, 1976).
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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