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.

中文翻译：

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有理约束和退化Lamé方程的静电问题

在本说明中，我们扩展了在平面静电场中由这些电荷以及一些固定点电荷产生的单位可移动点电荷的平衡构型与相应Lamé微分方程的多项式解之间的经典关系。即，我们发现受某种类型的有理或多项式约束的单位可动电荷的平衡构型与相应的退化Lamé方程的多项式解之间的相似关系，请参见下面的详细信息。特别地，经典Hermite和Laguerre多项式满足的标准线性微分方程属于此类。除了这两个经典情况外，我们还提供了许多其他示例，包括一些相对论正交多项式和它们满足的线性微分方程。