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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Interlacing and nonorthogonality of spectral polynomials for the Lamé operator
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by A. Bourget, T. McMillen and A. Vargas PDF
Proc. Amer. Math. Soc. 137 (2009), 1699-1710 Request permission

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
  • Received by editor(s): March 6, 2008
  • Published electronically: December 12, 2008
  • Communicated by: Andreas Seeger
  • © Copyright 2008 American Mathematical Society
  • 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
  • MathSciNet review: 2470828