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.