Electrostatic Problems with a Rational Constraint and Degenerate Lamé Equations
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In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lamé differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain type of rational or polynomial constraint and polynomial solutions of a corresponding degenerate Lamé equation, see details below. In particular, the standard linear differential equations satisfied by the classical Hermite and Laguerre polynomials belong to this class. Besides these two classical cases, we present a number of other examples including some relativistic orthogonal polynomials and linear differential equations satisfied by those.
KeywordsElectrostatic equilibrium Lamé differential equation
Mathematics Subject Classification (2010)Primary 31C10 33C45
The authors are sincerely grateful to the anonymous referees for their careful reading of the initial version of the manuscript and helpful suggestions. The second author is grateful to Universidade Estadual Paulista for the hospitality and excellent research atmosphere during his visit to São José do Rio Preto in July 2017.
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