A new approach to ratio asymptotics for orthogonal polynomials

Abstract

We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is uniquely determined by the normalization and leading coefficient. Several applications of this result are also discussed. One of our main theorems is that for regular measures on the closed unit disk - including, but not limited to the unit circle - one has ratio asymptotics along a sequence of asymptotic density 1.