The spectral analysis of three families of exceptional Laguerre polynomials

详细信息    查看全文

• 作者：Constanze Liaw^a ; ^{Constanze_Liaw@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; Lance L. Littlejohn^a ; ^{Lance_Littlejohn@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; Robert Milson^b ; ^{robert.milson@dal.ca" class="auth_mail" title="E-mail the corresponding author} ; Jessica Stewart^a ; ^{Jessica_Stewart@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 33C45 ; 34B24 ; 42C05 ; secondary ; 33C47 ; 47B25
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：5-41
• 全文大小：336 K

文摘

The Bochner Classification Theorem (1929) characterizes the polynomial sequences ${\left\{p_n\right\}}_{n=0}^{\infty }$, with degp_n=n$deg{p}_n=n$ 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, with certain restrictions on the polynomial parameters, satisfy these conditions. In 2009, Gómez-Ullate, Kamran, and Milson found that for sequences ${\left\{p_n\right\}}_{n=1}^{\infty }$, degp_n=n$deg{p}_n=n$ (without the constant polynomial), the only such sequences satisfying these conditions are the exceptional  X₁$X_1$-Laguerre and X₁$X_1$-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials {p_n}_n∈N0⧵A${\left\{p_n\right\}}_{n\in {\mathbb{N}}_0⧵A}$, where A$A$ is a finite subset of the non-negative integers N₀${\mathbb{N}}_0$ and where degp_n=n$deg{p}_n=n$ for all n∈N₀⧵A$n\in {\mathbb{N}}_0⧵A$. We call such a sequence an exceptional polynomial sequence of codimension |A|$\left|A\right|$, where the latter denotes the cardinality of A$A$. All exceptional sequences with a non singular weight, found to date, have the remarkable feature that they form a complete orthogonal set in their natural Hilbert space setting.

Among the exceptional sets already known are two types of exceptional Laguerre polynomials, called the Type I and Type II exceptional Laguerre polynomials, each omitting m$m$ polynomials. In this paper, we briefly discuss these polynomials and construct the self-adjoint operators generated by their corresponding second-order differential expressions in the appropriate Hilbert spaces. In addition, we present a novel derivation of the Type III family of exceptional 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 a complete spectral study of the second-order Type III exceptional Laguerre differential expression.