On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

In this paper we study the class of differential operators T = Pk j=1 QjD j with polynomial coefficients Qj in one complex variable satisfying the condition degQj ≤ j with equality for at least one j. We show that if degQk < k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n → ∞, as opposed to the case when degQk = k, which we have treated previously in [2]. Moreover we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of pn. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.