On the distribution and interlacing of the zeros of Stieltjes polynomials

Bourget, A.; McMillen, T.

doi:10.1090/S0002-9939-10-10348-7

On the distribution and interlacing of the zeros of Stieltjes polynomials
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by A. Bourget and T. McMillen PDF

Proc. Amer. Math. Soc. 138 (2010), 3267-3275 Request permission

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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