W. Van Assche

Orthogonal matrix polynomials and higher-order recurrence relations

Abstract It is well known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. We extend this result and show that every system of polynomials satisfying some (2 N +1)-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are N × N matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner product and other inner products in the linear space of polynomials. As an application we give a short proof of Kreins characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points.

Multiple orthogonal polynomials for classical weights

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to p > 1 weights satisfying Pearsons equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p + 1. We also obtain explicit formulas and recurrence relations for these polynomials.

Orthogonal polynomials with asymptotically periodic recurrence coefficients

Abstract Given the coefficients in the three term recurrence relation satisfied by orthogonal polynomials, we investigate the properties of those classes of polynomials whose coefficients are asymptotically periodic. Assuming a rate of convergence of the coefficients to their asymptotic values, we construct the measure with respect to which the polynomials are orthogonal and discuss their asymptotic behavior.

Some discrete multiple orthogonal polynomials

In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r + 1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r = 2.

RELATIVE ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO A DISCRETE SOBOLEV INNER PRODUCT

AbstractWe investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products

Multiple orthogonal polynomials associated with Macdonald functions

\left\langle {h,{\text{ }}g} \right\rangle = \int h g d\mu + \sum {_{j = 1}^m } \sum {_{i = 0}^{N_j } M_{j,i} h^{(i)} (c_j )} g^{(i)} (c_j )

EXTREMAL POLYNOMIALS ON DISCRETE SETS

, where μ is a certain type of complex measure on the real line, andcj are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure μ.

Information entropy of Gegenbauer polynomials

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Pade approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann-Hilbert problem, are presented. The first method uses a scalar Riemann-Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Rieman-Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.