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Proceedings of the American Mathematical Society

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Lamé differential equations and electrostatics

Authors: Dimitar K. Dimitrov and Walter Van Assche
Journal: Proc. Amer. Math. Soc. 128 (2000), 3621-3628
MSC (1991): Primary 34C10, 33C45; Secondary 34B30, 42C05
Published electronically: June 6, 2000
Erratum: Proc. Amer. Math. Soc. 131 (2003), 2303.
MathSciNet review: 1709747
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The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation \begin{equation*}A(x) y^{\prime\prime} + 2 B(x) y^{\prime} + C(x) y = 0, \end{equation*}where $A(x), B(x)$ and $C(x)$ are polynomials of degree $p+1, p$ and $p-1$, is under discussion. We concentrate on the case when $A(x)$ has only real zeros $a_{j}$ and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients $r_{j}$ in the partial fraction decomposition $B(x)/A(x) = \sum_{j=0}^{p} r_{j}/(x-a_{j})$, we allow the presence of both positive and negative coefficients $r_{j}$. The corresponding electrostatic interpretation of the zeros of the solution $y(x)$ as points of equilibrium in an electrostatic field generated by charges $r_{j}$ at $a_{j}$ is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.

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Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

Walter Van Assche
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Heverlee (Leuven), Belgium

Keywords: Lam\'e differential equation, electrostatic equilibrium, Laurent polynomials, Gegenbauer polynomials
Received by editor(s): February 22, 1999
Published electronically: June 6, 2000
Additional Notes: The research of the first author is supported by the Brazilian Science Foundations FAPESP under Grant 97/6280-0 and CNPq under Grant 300645/95-3.
The second author is a Research Director of the Belgian Fund for Scientific Research (FWO-V). Research supported by FWO research project G.0278.97.
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society