On a Two-Variable Class of Bernstein–Szegő Measures

Abstract

The one variable Bernstein-Szegő theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.

On the generalized Kesten–McKay distributions

“…Hence our results give substance and generalize the results of [4]. We will present a unified approach and recall and collect information on this family that is scattered though literature.…”

“…where, as agreed above, S j means in fact S (4) j (a 4 ). From this presentation it follows that may be the formula (1.16) is valid for m ≥ n − 1.…”

“…The fact that the polynomials P (n) m can be expressed as linear combination of the last at most n + 1 polynomials U m follows directly from [12] (Prop. 1 iii)) or from [4] (Lemma 3.1). Similar fact was noticed for n = 1, 2 earlier by Maroni in his papers published in the 90's.…”

“…It has to be noted that in 2009 there appeared paper [4]. Although the aim of it was to analyze two-dimensional measures on the plane of the form:…”

On linearly related orthogonal polynomials in several variables

“…Conversely, first of all, we are going to use an induction procedure to verify that {P n } n≥0 satisfies a three term relation as (1). Take the matrices A n,i , B n,i , and C n,i given by (11), (12), and (13), respectively. Multiplying (4) for n = 1 byà 0,i , it is easy to see that…”