Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials in kth-order SUSYQM

TL;DR

This paper extends the study of rationally-extended radial oscillator potentials and Laguerre exceptional orthogonal polynomials from second-order to kth-order supersymmetric quantum mechanics, deriving new identities and relations.

Contribution

It introduces a kth-order framework for rationally-extended radial oscillators and establishes differential relations linking exceptional orthogonal polynomials across orders.

Findings

Determined the polynomial degree in the potential denominator.

Derived first-order differential relations for exceptional polynomials.

Established identities connecting Laguerre polynomial products.

Abstract

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to $k$th-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a ($k-1$)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.