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

Consider an M‐th order linear differential operator, M≥2$M\ge 2$,L(M)=∑k=0Mρk(z)dkdzk,$$\begin{equation*} \mathcal {L}^{(M)}=\sum _{k=0}^{M}\rho _{k}(z)\frac{d^k}{dz^k}, \end{equation*}$$where ρM$\rho _M$ is a monic complex polynomial such that deg[ρM]=M$\mbox{{ \textrm {deg}}}[\rho _M ]=M$ and (ρk)k=0M−1$(\rho _k)_{k=0}^{M-1}$ are complex polynomials such that deg[ρk]≤k,0≤k≤M−1$\mbox{{ \textrm {deg}}}[\rho _k ]\le k, 0\le k \le M-1$. It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure μ. We obtain an asymptotic expansion of the eigenpolynomials of L(M)$\mathcal {L}^{(M)}$ in compact subsets out of the support of μ. 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]. [ABSTRACT FROM AUTHOR]