By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Tra...

For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and La...

This work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were construct...

Abstract. In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, ChristoffelDarboux formula and some ...

X iv :m at h/ 04 10 24 9v 1 [ m at h. C A ] 1 0 O ct 2 00 4 Some Systems of Multivariable Orthogonal Askey-Wilson Polynomials George Gasper* and Mizan Rahman† Abstract In 1991 Tratnik derived two systems of multivariable orthogonal Wilson polynomials and considered their limit cases. q-Analogues of these systems are derived, yielding systems of multivariable orthogonal Askey-Wilson polynomials ...

We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products 〈h, g〉 = ∫ hg dμ+ ∑m j=1 Nj i=0 Mj,ih (cj)g (cj), where μ is a certain type of complex measure on the real line, and cj are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corr...

We point out the relation between the orthogonal polynomials with respect to (w.r.t.) varying measures and the so-called orthogonal rationals on the unit circle in the complex plane. This observation enables us to combine different techniques in the study of these polynomials and rationals. As an example, we present a simple and short proof for a known result on the weak-star convergence of ort...

These lecture notes provide an introduction to orthogonal polynomials of several variables. It will cover the basic theory but deal mostly with examples, paying special attention to those orthogonal polynomials associated with classical type weight functions supported on the standard domains, for which fairly explicit formulae exist. There is little prerequisites for these lecture notes, a work...