Sobolev orthogonal polynomials have been studied extensively in the past 20 years. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.

a b s t r a c tIn this paper we state and prove some properties of the zeros of exceptional Jacobi and Laguerre polynomials. Generically, the zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between zeros of consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A Heine-Mehler type formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m. We also describe the location and the asymptotic behaviour of the m exceptional zeros, which converge for large n to fixed values.

No abstract

Given a matrix polynomial W(x), matrix bi-orthogonal polynomials with respect to the sesquilinear formwhere µ(x) is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to •, • W and matrix polynomials orthogonal with respect to µ(x) are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial W(x) we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of W(x). For perturbations with a singular leading coefficient several examples by Durán et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy. CONTENTS 1991 Mathematics Subject Classification. 42C05,15A23. Key words and phrases. Matrix orthogonal polynomials, Block Jacobi matrices, Darboux-Christoffel transformation, Block Cholesky decomposition, Block LU decomposition, quasi-determinants, non-Abelian Toda hierarchy. GA thanks financial support from the Universidad Complutense de Madrid Program "Ayudas para Becas y Contratos Complutenses Predoctorales en España 2011".MM & FM thanks financial support from the Spanish "Ministerio de Economía y Competitividad" research project MTM2012-36732-C03-01, Ortogonalidad y aproximación; teoría y aplicaciones.