Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III

TL;DR

This paper proves recurrence relations with constant coefficients for Laguerre and Jacobi multi-indexed orthogonal polynomials, extending bispectral techniques to derive explicit coefficients and exploring their bispectral properties.

Contribution

It provides the first rigorous proof of recurrence relations for Laguerre and Jacobi multi-indexed orthogonal polynomials and extends bispectral methods to these cases.

Findings

Proof of recurrence relations for Laguerre and Jacobi polynomials.

Explicit expressions for recurrence coefficients via bispectral techniques.

Extension of bispectral methods to new classes of orthogonal polynomials.

Abstract

In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by G\'omez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type.