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

Abstract

Consider an $M$-th order linear differential operator, $M\ge 2$, $${\mathcal{L}}^{(M)}=\sum _{k=0}^M{\rho }_k(z)\frac{d^k}{d{z}^k},$$ where ${\rho }_M$ is a monic complex polynomial such that $degree[{\rho }_M]=M$ and $({\rho }_k{)}_{k=0}^{M-1}$ are complex polynomials such that $degree[{\rho }_k]\le k,0\le k\le M-1$. 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 ${\mathcal{L}}^{(M)}$ in compact subsets out the support of $\mu$. In particular, we solve a conjecture posed in G.~Masson and B.~Shapiro, ``On polynomial eigenfunctions of a hypergeometric type operator,'' Exper. Math., vol.~10, pp.~609--618, 2001.