A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

Abstract

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula Rn+1(z)=[(1+icn+1)z+(1−icn+1)]Rn(z)−4dn+1zRn−1(z),n≥1, with R0(z)=1 and R1(z)=(1+ic1)z+(1−ic1), where {cn}n=1∞ is a real sequence and {dn}n=1∞ is a positive chain sequence. We establish that there exists a unique nontrivial probability measure μ on the unit circle for which {Rn(z)−2(1−mn)Rn−1(z)} gives the sequence of orthogonal polynomials. Here, {mn}n=0∞ is the minimal parameter sequence of the positive chain sequence {dn}n=1∞. The element d1 of the chain sequence, which does not affect the polynomials Rn, has an influence in the derived probability measure μ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {Mn}n=0∞ is the maximal parameter sequence of the chain sequence, then the measure μ is such that M0 is the size of its mass at z=1. An example is also provided to completely illustrate the results obtained.