Two variable deformations of the Chebyshev measure

Abstract

We construct one and two parameter deformations of the two dimensional Chebyshev polynomials with simple recurrence coefficients, following the algorithm in [3]. Using inverse scattering techniques, we compute the corresponding orthogonality measures.

On a Two-Variable Class of Bernstein–Szegő Measures

“…give a complete set of orthogonal polynomials in the total degree ordering. It follows that the recurrence coefficients in the total degree ordering are as suggested in [5], i.e. A x,n and A y,n are the same as in the previous example.…”

“…The polynomials and the recurrence coefficients in the lexicographical ordering are given in [5] using a Darboux transformation connecting the two examples.…”

“…Orthonormal polynomials in the total degree ordering as well as the lexicographical and reverse lexicographical orderings are considered. In section 4 we investigate the properties of the coefficients in the recurrence formulas satisfied by these polynomials and in section 5 we consider several examples including the ones discussed in [5].…”

“…In this article we will consider a special class of "Bernstein-Szegő" measures (see below for the definition) which are generalizations of the examples considered in [5] and construct orthogonal polynomials where the orderings used will be the total degree ordering or the lexicographical ordering. We will do this in order to investigate and compare the properties of the polynomials constructed.…”

The Darboux process and a noncommutative bispectral problem: some explorations and challenges

“…In this section we mention that in [G9] one finds a specific random walk introduced by Hoare and Rahman, see [HR] which we show leads to a bispectral situation in terms of polynomials of two variables. I also want to mention that in the multivariable case one finds a version of the Darboux process to obtain interesting deformations of the two dimensional Chebyshev measure, see [GI1].…”