The spectral analysis of three families of exceptional Laguerre polynomials

• Published in: Journal of approximation theory 2016-02, Vol.202, p.5-41
• Main Authors: Liaw, Constanze, Littlejohn, Lance L., Milson, Robert, Stewart, Jessica
• Format: Article
• Language: English
• Subjects: Orthogonal polynomials; Root asymptotics; Spectral theory
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2015.11.001

The Bochner Classification Theorem (1929) characterizes the polynomial sequences {pn}n=0∞, with degpn=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 {pn}n=1∞, degpn=n (without the constant polynomial), the only such sequences satisfying these conditions are the exceptionalX1-Laguerre and X1-Jacobi polynomials. Subsequently, during the past five years, several mathematicians and physicists have discovered and studied other exceptional orthogonal polynomials {pn}n∈N0⧵A, where A is a finite subset of the non-negative integers N0 and where degpn=n for all n∈N0⧵A. We call such a sequence an exceptional polynomial sequence of codimension |A|, where the latter denotes the cardinality of 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 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.