Local asymptotics for orthonormal polynomials on the unit circle via universality

Let µ be a positive measure on the unit circle that is regular in the sense of Stahl, Totik, and Ullmann. Assume that in some subarc J, µ is absolutely continuous, while µ ′ is positive and continuous. Let { φ n } be the orthonormal polynomials for µ . We show that for appropriate ζ n ∈ J , { φ n ( ζ n ( 1 + z n ) ) φ n ( ζ n ) } n ≥ 1 is a normal family in compact subsets of ℂ. Using universality limits, we show that limits of subsequences have the form e z + C ( e z − 1) for some constant C . Under additional conditions, we can set C = 0.