On Higher Heine-stieltjes Polynomials

Take a linear ordinary differential operator d(z) = Pk i=1 Qi(z) d dzi with polynomial coefficients and set r = maxi=1,...,k(degQi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) degQk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [6] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has a polynomial solution S(z) of degree n. We have shown that under some mild non-degeneracy assumptions on T there exist exactly ` n+r n ́ spectral polynomials Vn,i(z) of degree r and their corresponding eigenpolynomials Sn,i(z) of degree n. Localization results of [6] provide the existence of abundance of converging as n → ∞ sequences of normalized spectral polynomials {e Vn,in (z)} where e Vn,in (z) is the monic polynomial proportional to Vn,in (z). Below we calculate for any such converging sequence {e Vn,in (z)} the asymptotic rootcounting measure of the corresponding family {Sn,in (z)} of eigenpolynomials. We also conjecture that the sequence of sets of all normalized spectral polynomials {e Vn,i(z)} having eigenpolynomials S(z) of degree n converges as n → ∞ to the standard measure in the space of monic polynomials of degree r which depends only on the leading coefficient Qk(z).