Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences

Abstract: It was shown recently that associated with a pair of real sequences ${{c_{n}}{n=1}^{\infty}, {d{n}}{n=1}^{\infty}}$, with ${d{n}}{n=1}^{\infty}$ a positive chain sequence, there exists a unique nontrivial probability measure $\mu$ on the unit circle. The Verblunsky coefficients ${\alpha{n}}{n=0}^{\infty}$ associated with the orthogonal polynomials with respect to $\mu$ are given by the relation $$ \alpha{n-1}=\overline{\tau}{n-1}\left[\frac{1-2m{n}-ic_{n}}{1-ic_{n}}\right], \quad n \geq 1, $$ where $\tau_0 = 1$, $\tau_{n}=\prod_{k=1}^{n}(1-ic_{k})/(1+ic_{k})$, $n \geq 1$ and ${m_{n}}{n=0}^{\infty}$ is the minimal parameter sequence of ${d{n}}{n=1}^{\infty}$. In this manuscript we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences ${c{n}}{n=1}^{\infty}$ and ${m{n}}{n=1}^{\infty}$. When the sequence $ {c{n}}{n=1}^{\infty}$ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of $z= -1$. Furthermore, we show that it is possible to ge-nerate periodic Verblunsky coefficients by choosing periodic sequences ${c{n}}{n=1}^{\infty}$ and ${m{n}}{n=1}^{\infty}$ with the additional restriction $c{2n}=-c_{2n-1}, \, n\geq 1.$ We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.