一类新的例外正交多项式：III型$X_{m}$ -Laguerre多项式及三类例外Laguerre多项式的谱分析
A New Class of Exceptional Orthogonal Polynomials: The Type III $X_{m}$-Laguerre Polynomials And The Spectral Analysis of Three Types of Exceptional Laguerre Polynomials
摘要：
Bochner分类定理（1929）描述了多项式序列$\p_{n}\}_{n=0}^{\infty}$，其中$\text{deg}\,p_{n}=n$同时形成二阶微分算子的本征态的完整集合，并且关于具有所有阶的有限矩的正Borel测度正交。事实上，直到一个复杂的线性变化的变量，只有经典的埃尔米特，拉盖尔，雅可比多项式满足这些条件。在2009年，戈麦斯-尤拉特、卡姆兰和米尔森发现，对于序列$\{p_{n}\}_{n=1}^{\infty}$、$\text{deg}\,p_{n}=n$（没有常数多项式），唯一的这样的序列是例外的$X_{1}$ -拉盖尔和$X_{1}$ -雅可比多项式。随后，发现并研究了其他例外正交多项式$\{p_{n}\}_{n\in\mathbb{N}_{0}\diagdown A}$（这里$A$是所有$n\in\mathbb{N}_{0}\diagdown A$的非负整数$\mathbb{N}_{0}$和$\text{deg}\,p_{n}=n$的有限子集）。我们称这样的序列为例外$X_{\left\vert A\right\vert}$序列。值得注意的是，迄今为止发现的所有例外序列在其自然希尔伯特空间设置中形成完全正交集。在已知的例外集合中有I型和II型$X_{m}$ -拉盖尔多项式，每个都省略了$m$多项式。我们简要地讨论了这些多项式，并在适当的Hilbert空间中构造了由其相应的二阶微分表达式生成的自伴算子。此外，我们提出了一个新的$X_{m}$-Laguerre多项式族，并详细讨论了它的性质。我们包括这些多项式的几个表示，正交性，规范，完整性，它们的局部极值和根的位置，根渐近，以及二阶III型例外$X_{m}$ -Laguerre微分表达式的谱研究。
英文摘要：
The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\p_{n}\}_{n=0}^{\infty}$, with $\text{deg}\,p_{n}=n$ that simultaneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders. Indeed, up to a complex linear change of variable, only the classical Hermite, Laguerre, and Jacobi polynomials satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences $\{p_{n}\}_{n=1}^{\infty}$, $\text{deg}\,p_{n}=n$ (without the constant polynomial), the only such sequences are the exceptional $X_{1}$-Laguerre and $X_{1}$-Jacobi polynomials. Subsequently, other exceptional orthogonal polynomials $\{p_{n}\}_{n\in\mathbb{N}_{0}\diagdown A}$ were discovered and studied (here $A$ is a finite subset of the non-negative integers $\mathbb{N}_{0}$ and $\text{deg}\,p_{n}=n$ for all $n\in\mathbb{N}_{0}\diagdown A$). We call such a sequence an exceptional $X_{\left\vert A\right\vert}$ sequence. Remarkably, all exceptional sequences found, to date, form a complete orthogonal set in their natural Hilbert space setting. Among the exceptional sets already known are the Type I and Type II $X_{m}$-Laguerre polynomials, each omitting $m$ polynomials. We briefly discuss these polynomials and construct self-adjoint operators generated by their corresponding second-order differential expressions in appropriate Hilbert spaces. In addition, we present a new Type III family of $X_{m}$-Laguerre polynomials along with a detailed disquisition of its properties. We include several representations of these polynomials, orthogonality, norms, completeness, the location of their local extrema and roots, root asymptotics, as well as the spectral study of the second-order Type III exceptional $X_{m}$-Laguerre differential expression.
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