Electrostatic problems with a rational constraint and degenerate Lame equations

TL;DR

This paper extends the classical link between electrostatic equilibrium configurations and polynomial solutions of Lamé equations to include rational constraints and degenerate cases, encompassing classical and relativistic orthogonal polynomials.

Contribution

It introduces a new relation connecting equilibrium configurations of charges with polynomial solutions of degenerate Lamé equations under rational constraints, broadening classical results.

Findings

Established relation for rational constraints and degenerate Lamé equations.

Included classical Hermite and Laguerre polynomials as special cases.

Presented examples involving relativistic orthogonal polynomials.

Abstract

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 Lam\'e 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 Lam\'e 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.