On the orthogonality of the derivative of the reciprocal sequence

摘要

Let {Φn} 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 Φn(z)=zn.A similar problem can be posed in terms of the reciprocal sequence of {Φn} as follows:If Φn+1(0)≠0, we can define the monic sequence {Pn} byPn(z)=(Φn+1∗)′(z)(n+1)Φn+1(0)n∈N={0,1,…},where Φn∗ denotes the reciprocal polynomial of Φn, and to study their orthogonality conditions.In this paper we obtain a necessary and sufficient condition for the regularity of {Pn} when the first reflection coefficient Φ1(0) is a real number. Also, we give an explicit representation for {Φn} and {Pn}.Moreover, we analyse some questions concerning to the associated functionals of them sequences and the positive definite and semiclassical character.