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 considered, which leads to an equilibrium with a non-convex external field.