arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1909.13752math.CA
收藏

关于与Hahn算子有关的经典正交多项式

On classical orthogonal polynomials related to Hahn's operator

R. Álvarez-Nodarse, K. Castillo, D. Mbouna, J. Petronilho

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设${\bf u}$是作用在多项式空间上的非零线性泛函。设$\mathbf{D}_{q,ω}$是作用在多项式对偶空间上的Hahn算子.假设存在多项式$ϕ$和$ψ$，以及$\mathrm{deg}\,ϕ\leq2$和$\mathrm{deg}\,ψ\leq1$，使得函数方程$$ \mathbf{D}_{q,ω}(ϕ{\bf u})=ψ{\bf u} $$成立，其中所涉及的操作在分布意义上定义。本文给出了${\bf u}$正则的充要条件，即存在关于${\bf u}$的一列正交多项式，这些条件只涉及$ϕ$和$ψ$的系数.证明中的一个关键步骤依赖于这样一个事实，即分布Rodrigues型公式在不假设${\bf u}$是正则的情况下成立。

英文摘要：

Let ${\bf u}$ be a nonzero linear functional acting on the space of polynomials. Let $\mathbf{D}_{q,ω}$ be a Hahn operator acting on the dual space of polynomials. Suppose that there exist polynomials $ϕ$ and $ψ$, with $\mathrm{deg}\,ϕ\leq2$ and $\mathrm{deg}\,ψ\leq1$, so that the functional equation $$ \mathbf{D}_{q,ω}(ϕ{\bf u})=ψ{\bf u} $$ holds, where the involved operations are defined in a distributional sense. In this note we state necessary and sufficient conditions, involving only the coefficients of $ϕ$ and $ψ$, such that ${\bf u}$ is regular, that is, there exists a sequence of orthogonal polynomials with respect to ${\bf u}$. A key step in the proof relies upon the fact that a distributional Rodrigues-type formula holds without assuming that ${\bf u}$ is regular.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑