On the orthogonality of the derivative of the reciprocal sequence

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 { Φ′ n+1 /( n+1)} 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 P n(z)= (Φ n+1 ∗)′(z) (n+1) Φ n+1(0) n∈ N={0,1,…}, where Φ n ∗ denotes the reciprocal polynomial of Φ n , and to study their orthogonality conditions. In this paper we obtain a necessary and sufficient condition for the regularity of { P n } when the first reflection coefficient Φ 1(0) is a real number. Also, we give an explicit representation for { Φ n } and { P n }. Moreover, we analyse some questions concerning to the associated functionals of them sequences and the positive definite and semiclassical character.