An asymptotic expansion of eigenpolynomials for a class of linear differential operators

Consider an M-th order linear differential operator, M >= 2, L-(M) =Sigma(M)(k=0) rho(k) (z) d(k) / dz(k), where rho(k) is a monic complex polynomial such that deg[rho(M)] = M and (rho(k))(k=0)(M-1) are complex polynomials such that deg[rho(k)] <= k, 0 <= k <= M-1. It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure mu. We obtain an asymptotic expansion of the eigenpolynomials of L-(M) in compact subsets out of the support of mu. In particular, we solve a conjecture posed in Masson and Shapiro [On polynomial eigenfunctions of a hypergeometric type operator. Exper Math. 2001;10:609-618].