Lamé Differential Equations and Electrostatics

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A(x)y′′ + 2B(x)y′ + C(x)y = 0, 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 aj and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients rj in the partial fraction decomposition B(x)/A(x) = ∑p j=0 rj/(x − aj), we allow the presence of both positive and negative coefficients rj . The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges rj at aj 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.