Lamé differential equations and electrostatics

Abstract

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A ( x ) y ′ ′ + 2 B ( x ) y ′ + C ( x ) y = 0 , \begin{equation*} A(x) y^{\prime \prime } + 2 B(x) y’ + C(x) y = 0, \end{equation*} where A ( x ) , B ( x ) A(x), B(x) and C ( x ) C(x) are polynomials of degree p + 1 , p p+1, p and p − 1 p-1 , is under discussion. We concentrate on the case when A ( x ) A(x) has only real zeros a j a_{j} and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients r j r_{j} in the partial fraction decomposition B ( x ) / A ( x ) = ∑ j = 0 p r j / ( x − a j ) 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 r_{j} . The corresponding electrostatic interpretation of the zeros of the solution y ( x ) y(x) as points of equilibrium in an electrostatic field generated by charges r j r_{j} at a j 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.