We consider polynomials that are orthogonal on [−1, 1] with respect to a modified Jacobi weight (1 − x) α (1 + x) β h(x), with α, β > −1 and h real analytic and stricly positive on [−1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1, 1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1, 1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szegő function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.

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dedicated to jaap korevaar on his 75th birthdayWe study the zeros of orthogonal polynomials p n, N , n=0, 1, ..., that are generated by recurrence coefficients a n, N and b n, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio nÂN converges, we show that the polynomials p n, N have an asymptotic zero distribution as nÂN tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distributions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes Wigert polynomials, and Lommel polynomials.

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 Pearson's 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.

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. In this paper we extend this result and show that every system of polynomials satisfying some (2N + 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 Krein’s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points.Belgian National Fund for Scientific Researc

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

Abstract. 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 Pearson's 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.

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