Methods and Applications of Analysis

Volume 22 (2015)

Number 3

$-1$ Krall–Jacobi polynomials

Pages: 249 – 258

DOI: http://dx.doi.org/10.4310/MAA.2015.v22.n3.a1

Authors

Luc Vinet (Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada)

Guo-Fu Yu (Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Alexei Zhedanov (Donetsk Institute for Physics and Technology, Donetsk, Ukraine)

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 $q = -1$.

Keywords

Jacobi polynomials, little $q$-Jacobi polynomials, Geronimus transformation

2010 Mathematics Subject Classification

33C45, 33C47, 42C05

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