A new class of orthogonal polynomials is introduced which generalizes the Bernstein-Szegö polynomials and includes the associated polynomials as well. The purpose of this paper is to give a natural extension of the Bernstein-Szegö orthogonal polynomials for a general class of weight functions. A nonnegative function w defined on the real line is called a weight function if w > 0, fRw > 0 and al...

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szegő recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the arc { e : α ≤ θ ≤ 2π−α } where cos α 2 def = √ 1− |a|2 with α ∈ (0, π). We analyze the orthogonal polynomials by comparing th...

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relation...

A new set of special functions is described which has a wide range of applications, from number theory to integrability of non-linear dynamical systems. 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 t...

Multiple orthogonal polynomials are polynomials in one variable that satisfy orthogonality conditions with respect to several measures. I will briefly give some general properties of these polynomials (recurrence relation, zeros, etc.). These polynomials have recently appeared in many applications, such as number theory, random matrices, non-intersecting random paths, integrable systems, etc. I...

We study polynomials which satisfy the same recurrence relation as the Szegő polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szegő polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szegő polynomials, para-orthogonal ...

We consider a modiication of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x = 0. We obtain the resulting orthogonal polynomials, identify them as hypergeometric functions and derive the second order diierence equation which these polynomials satisfy. The corresponding tridiagonal matrices and associated polynomials were also studied. x1 Intr...

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

Abstract. We find a local (d + 1)× (d + 1) Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree d. Our Riemann-Hilbert problem is similar to a local d × d RiemannHilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polyno...