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.