Interlacing and nonorthogonality of spectral polynomials for the Lamé operator

Bourget, A.; McMillen, T.; Vargas, A.

doi:10.1090/S0002-9939-08-09811-0

Interlacing and nonorthogonality of spectral polynomials for the Lamé operator
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by A. Bourget, T. McMillen and A. Vargas

Proc. Amer. Math. Soc. 137 (2009), 1699-1710

DOI: https://doi.org/10.1090/S0002-9939-08-09811-0

Published electronically: December 12, 2008

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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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