Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials

TL;DR

This paper explores the mathematical properties of the recently discovered exceptional $X_{ ext{ell}}$ Laguerre and Jacobi polynomials, including their differential equations, factorization, and recurrence relations.

Contribution

It provides new simplified forms and detailed analysis of the properties of $X_{ ext{ell}}$ polynomials, expanding understanding of their structure and relations.

Findings

Derived simpler forms of $X_{ ext{ell}}$ polynomials

Analyzed factorization and shape invariance of differential operators

Established recurrence relations and generating functions

Abstract

We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the $X_{\ell}$ polynomials.