Note on the X_(1)-Laguerre orthogonal polynomials

TL;DR

This paper provides additional insights into the structure and spectral properties of X_(1)-Laguerre orthogonal polynomials, focusing on their differential equations and self-adjoint operators within a weighted Hilbert space.

Contribution

It offers new details on the Sturm-Liouville form and boundary conditions that define the self-adjoint operator associated with X_(1)-Laguerre polynomials.

Findings

Characterization of the differential operator in Sturm-Liouville form

Specification of boundary conditions for self-adjointness

Identification of the discrete spectrum and eigenvectors

Abstract

This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the special self-adjoint operator, whose discrete spectrum and associated eigenvectors yield the X_(1)-Laguerre polynomials.