An asymptotic expansion of eigenpolynomials for a class of linear differential operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917603823321088 |
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| author | Borrego-Morell, Jorge A. |
| author_facet | Borrego-Morell, Jorge A. |
| contents | Consider an $M$-th order linear differential operator, $M\geq 2$, $$ \mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $ρ_M $ is a monic complex polynomial such that $degree[ρ_M]=M$ and $(ρ_k)_{k=0}^{M-1}$ are complex polynomials such that $degree[ ρ_k ]\leq k, 0\leq k \leq M-1$. It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure $μ$. We obtain an asymptotic expansion of the eigenpolynomials of $\mathcal{L}^{(M)}$ in compact subsets out the support of $μ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An asymptotic expansion of eigenpolynomials for a class of linear differential operators Borrego-Morell, Jorge A. Classical Analysis and ODEs 41A60, 30E15, 34E05, 34E10, 34E20, 34L20, 34M30 Consider an $M$-th order linear differential operator, $M\geq 2$, $$ \mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $ρ_M $ is a monic complex polynomial such that $degree[ρ_M]=M$ and $(ρ_k)_{k=0}^{M-1}$ are complex polynomials such that $degree[ ρ_k ]\leq k, 0\leq k \leq M-1$. It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure $μ$. We obtain an asymptotic expansion of the eigenpolynomials of $\mathcal{L}^{(M)}$ in compact subsets out the support of $μ$. 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. |
| title | An asymptotic expansion of eigenpolynomials for a class of linear differential operators |
| topic | Classical Analysis and ODEs 41A60, 30E15, 34E05, 34E10, 34E20, 34L20, 34M30 |
| url | https://arxiv.org/abs/2403.02007 |