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 {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {αn}n=0∞ associated with the orthogonal polynomials with respect to μ are given by the relation αn-1=τ¯n-1[1-2mn-icn1-icn],n≥1,where τ0= 1 , τn=∏k=1n(1-ick)/(1+ick), n≥ 1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {cn}n=1∞ and {mn}n=1∞. When the sequence {cn}n=1∞ 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 {cn}n=1∞ and {mn}n=1∞ with the additional restriction c2n=-c2n-1,n≥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.