On the orthogonality of the derivative of the reciprocal sequence (II)

Abstract

Let {Φ n } be a monic orthogonal polynomial sequence on the unit circle (MOPS). The study of the orthogonality properties of the derivative sequence is a classic problem of the orthogonal polynomials theory. In fact, it is well known that the derivative sequence is again a MOPS if and only if Φ n (z) = z n . A similar problem can be posed in terms of the reciprocal sequence of {Φ n } as follows: If Φ n+1(0) ≠ 0, we can define the monic sequence {P n } by: where denotes the reciprocal polynomial of Φ n , and to study their orthogonality conditions. In this paper, we give a explicit representation for the orthogonal sequences {Φ n } and {P n }. In this way, we obtain concrete examples of families of orthogonal polynomials on the unit circle, as well as studying some properties about the differential behavoir of orthogonal polynomials on the unit circle.