论文标题
特殊的Legendre多项式和汇合Darboux转换
Exceptional Legendre Polynomials and Confluent Darboux Transformations
论文作者
论文摘要
特殊的正交多项式是正交多项式的家族，它们是Sturm-Liouville特征值问题的解决方案。他们通过允许错过有限数量的“异常”度的多项式序列来概括Hermite，Laguerre和Jacobi多项式的古典家族。在本文中，我们通过考虑经典Legendre操作员的同一光谱变形，介绍了多参数杰出的多项式多项式的新结构。使用汇合的darboux变换和逆散射理论的技术，我们获得了对操作员和多项式的完全明确描述。本文的主要新颖性是新颖的结构，它允许具有任意数量的实际参数的特殊多项式家庭。
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.
