Methods and Applications of Analysis
Volume 22 (2015)
$-1$ Krall–Jacobi polynomials
Pages: 249 – 258
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 $q = -1$.
Jacobi polynomials, little $q$-Jacobi polynomials, Geronimus transformation
2010 Mathematics Subject Classification
33C45, 33C47, 42C05