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Electrostatic interpretation of zeros of orthogonal polynomials


Author: Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 146 (2018), 5323-5331
MSC (2010): Primary 31C45, 33C45, 34B24, 34C10, 34E99
DOI: https://doi.org/10.1090/proc/14226
Published electronically: September 17, 2018
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Abstract: We study the differential equation $ - (p(x) y')' + q(x) y' = \lambda y,$ where $ p(x)$ is a polynomial of degree at most 2 and $ q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials, and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: a set of distinct, real numbers $ \left \{x_1, \dots , x_n\right \}$ satisfies

$\displaystyle p(x_i) \sum _{k = 1 \atop k \neq i}^{n}{\frac {2}{x_k - x_i}} = q(x_i) - p'(x_i) \qquad \textup {for all}~ 1\leq i \leq n$    

if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on $ p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.

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Additional Information

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/14226
Keywords: Orthogonal polynomials, zeros, electrostatic interpretation.
Received by editor(s): April 25, 2018
Received by editor(s) in revised form: April 26, 2018
Published electronically: September 17, 2018
Communicated by: Mourad Ismail
Article copyright: © Copyright 2018 American Mathematical Society

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