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On the distribution and interlacing of the zeros of Stieltjes polynomials


Authors: A. Bourget and T. McMillen
Journal: Proc. Amer. Math. Soc. 138 (2010), 3267-3275
MSC (2010): Primary 34L20, 34B30
DOI: https://doi.org/10.1090/S0002-9939-10-10348-7
Published electronically: March 24, 2010
MathSciNet review: 2653956
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Abstract: Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of certain Stieltjes polynomials of successive degrees interlace.


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Additional Information

A. Bourget
Affiliation: Department of Mathematics, McCarthy Hall 154, California State University at Fullerton, Fullerton, California 92834
Email: abourget@fullerton.edu

T. McMillen
Affiliation: Department of Mathematics, McCarthy Hall 154, California State University at Fullerton, Fullerton, California 92834
Email: tmcmillen@fullerton.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10348-7
Keywords: Lam\'e equation, interlacing zeros, Heine-Stieltjes polynomials, Van Vleck polynomials, orthogonal polynomials
Received by editor(s): August 10, 2009
Received by editor(s) in revised form: November 25, 2009, and December 19, 2009
Published electronically: March 24, 2010
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.