Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

Abstract

We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second-order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum-mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

Exceptional Gegenbauer polynomials via isospectral deformation

“…Shortly after that, Durán has shown that there is an alternative way to construct deformed OPs, by first perturbing the measure of the discrete Hahn polynomials, dualizing and taking a suitable limit. 31 Durán's work implies that the most general class of XOPs contains not just deformations of classical OPs, but deformations of other XOPs. He did this by exhibiting constructions and examples where setting the deformation parameters to zero recovers exceptional, rather than classical OPs.…”

“…Finally, Ref. 31 is significant because of explicit examples of alternative constructions of deformable Legendre polynomials via DDTs of classical Jacobi operators with negative integer parameters.…”

Exceptional Gegenbauer polynomials via isospectral deformation

“…The family of exceptional Legendre polynomials of the second kind was recently introduced by some of the present authors [13], showing for the first time that deformations of classical polynomials with an arbitrary number of real parameters exist. Shortly after that, Durán has shown that there is an alternative way to construct exceptional polynomials of the second kind, by first perturbing the measure of the discrete Hahn polynomials, dualizing and taking a suitable limit [11]. The extra continuous parameters enter in this construction in the polynomial perturbation of the classical discrete measure.…”