Note on the X_(1)-Jacobi orthogonal polynomials

TL;DR

This paper provides additional insights into the structure, boundary conditions, and spectral properties of the X_(1)-Jacobi orthogonal polynomials, complementing previous work on their differential equations and orthogonality.

Contribution

It offers detailed analysis of the Sturm-Liouville form and boundary conditions for the X_(1)-Jacobi polynomials, enhancing understanding of their spectral properties.

Findings

Derived the Sturm-Liouville form of the differential equation

Specified boundary conditions for the self-adjoint operator

Characterized the discrete spectrum and eigenvectors

Abstract

This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) 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)-Jacobi polynomials.