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Interlacing and nonorthogonality of spectral polynomials for the Lamé operator


Authors: A. Bourget, T. McMillen and A. Vargas
Journal: Proc. Amer. Math. Soc. 137 (2009), 1699-1710
MSC (2000): Primary 34L05, 34B30
DOI: https://doi.org/10.1090/S0002-9939-08-09811-0
Published electronically: December 12, 2008
MathSciNet review: 2470828
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Abstract:

Polynomial solutions to the Heine-Stieltjes equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's in various contexts including the solution of the Laplace equation on an ellipsoid. Recently there has been renewed interest in the distribution of the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials increases. In this paper we show that the zeros of Van Vleck polynomials corresponding to Stieltjes polynomials of successive degrees interlace. We also show that the spectral polynomials formed from the Van Vleck zeros are not orthogonal with respect to any measure. This furnishes a counterexample, coming from a second order differential equation, to the converse of the well-known theorem that the zeros of orthogonal polynomials interlace.


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Additional Information

A. Bourget
Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92834
Email: abourget@fullerton.edu

T. McMillen
Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92834
Email: tmcmillen@fullerton.edu

A. Vargas
Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92834
Email: tvargas@csu.fullerton.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09811-0
Received by editor(s): March 6, 2008
Published electronically: December 12, 2008
Communicated by: Andreas Seeger
Article copyright: © Copyright 2008 American Mathematical Society

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