Asymptotics for orthogonal polynomials and separation of their zeros

Let p_n denote the orthonormal polynomials associated with a measure μ with compact support on the real line. In a recent paper, we showed there is a close relationship between the spacing of zeros of successive orthogonal polynomials p_n,p_n−1, and uniform bounds on the orthogonal polynomials in subintervals of the support. In this paper, we show there is also a relationship between asymptotics for the spacing of zeros of p_n−1,p_n, and pointwise asymptotics for the orthogonal polynomials.