A special class of polynomials orthogonal on the unit circle including the associated polynomials

Abstract

Let (P _ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ω_υ) the monic associated polynomials of (P _v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (AP_υλ+BΩ_υλ)/A^λ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP _ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP _ν and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.