A Christoffel–Darboux formula and a Favard's theorem for orthogonal Laurent polynomials on the unit circle

A matrix approach to the computation of quadrature formulas on the unit circle

“…and p(n) = E n 2 respectively, where as usual, E[x] denotes the integer part of x (see [3], [9] and [10] for other properties for these particular orderings). In the sequel we will denote by {φ n (z)} ∞ n=0 the sequence of monic orthogonal Laurent polynomials for the measure µ and the generating sequence {p(n)} ∞ n=0 , that is, when the leading coefficients are equal to 1 for all n ≥ 0 (coefficients of z q(n) or z −p(n) when s(n) = 0 or s(n) = 1 respectively).…”

“…An alternative proof is given in [10] making use of (4) and Proposition 2.2. We will see now that the recurrences (4) and (1) (the same with (2)) are in fact equivalent.…”

“…Consider, for n ≥ 2, the eight cases (s(n), s(n − 1), s(n − 2)) = (i, j, k) with i, j, k ∈ {0, 1}. As we said, the proofs in the balanced situations (0, 1, 0) and (1, 0, 1) were given in [10] making use of Proposition 2.2 and the Szegő recurrence (4). Proceeding in the same way in the remaining cases we obtain:…”

“…Here, it should be remarked that the theory of orthogonal Laurent polynomials on the unit circle establishes features totally different to the theory on the real line because of the close relation between orthogonal Laurent polynomials and the orthogonal polynomials on the unit circle (see [9]). …”

“…p(n) = 0 for all n) is given in [26] and an alternative simpler approach in [15] (see also [1]). A proof based on the techniques introduced in the Chihara's book [7] in the balanced situations p(n) = E n+1 2 and p(n) = E n 2 is given in [9]. Anyway, the proof presented here based upon the above equivalences between recurrences is very much simpler, and still valid when a general generating sequence is considered.…”