-1 Krall-Jacobi Polynomials

TL;DR

This paper introduces a new family of Krall-Jacobi polynomials with a point mass inside their orthogonality interval, derived from little -1 Jacobi polynomials via a Geronimus transform, involving a third order Dunkl-type differential operator.

Contribution

It presents the first nontrivial example of Krall-type polynomials with an internal point mass, expanding the theory of orthogonal polynomials with modified measures.

Findings

First nontrivial Krall-type polynomials with an internal mass

Polynomials satisfy a third order Dunkl-type eigenvalue equation

Derived from little -1 Jacobi polynomials via Geronimus transform

Abstract

We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the little $q$-Jacobi polynomials in the limit $q=-1$.