1. An asymptotic expansion of eigenpolynomials for a class of linear differential operators
- Author
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Borrego-Morell, Jorge A.
- Subjects
Mathematics - Classical Analysis and ODEs ,41A60, 30E15, 34E05, 34E10, 34E20, 34L20, 34M30 - Abstract
Consider an $M$-th order linear differential operator, $M\geq 2$, $$ \mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ 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 $degree[ \rho_k ]\leq k, 0\leq k \leq M-1$. 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.
- Published
- 2024
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