Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials

Interlacing and asymptotic properties of Stieltjes polynomials

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacin...

Asymptotic Properties of Heine-Stieltjes and Van Vleck Polynomials

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of non-positive charges is consider...

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 i...

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated w...

On the average number of sharp crossings of certain Gaussian random polynomials