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

$$\begin{aligned} \alpha _{n-1}=\overline{\tau }_{n-1}\left[ \frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right] , \quad n \ge 1, \end{aligned}$$

where $\tau _0 = 1$, $\tau _{n}=\prod _{k=1}^{n}(1-ic_{k})/(1+ic_{k})$, $n \ge 1$ 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 generate periodic Verblunsky coefficients by choosing periodic sequences $\{c_{n}\}_{n=1}^{\infty }$ and $\{m_{n}\}_{n=1}^{\infty }$ with the additional restriction $c_{2n}=-c_{2n-1}, \, n\ge 1.$ 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.