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We show that universality limits and bounds for orthonormal polynomials imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if μ is a regular measure supported on [−1, 1], and in a neighborhood of 1, μ is absolutely continuous, while for some α > −1, μ′ (t) = h (t) (1− t), where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic lim n→∞ pn ( 1− z 2 2n2 ) pn (1) = J∗ α (z) J∗ α (0) uniformly in compact subsets of the plane. Here J∗ α (z) = Jα (z) /z α is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics. 1. Results Let μ be a finite positive Borel measure with compact support, contianing infinitely many points. Then we may define orthonormal polynomials pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... satisfying the orthonormality conditions ∫ pnpmdμ = δmn. We denote the zeros of pn by xnn < xn−1,n < ... < x2n < x1n. The {pn} satisfy the three term recurrence relation xpn−1 (x) = anpn (x) + bn−pn− (x) + an−1pn−2 (x) , where an = γn−1 γn and bn ∈ R. Asymptotics for pn as n → ∞ are a much studied subject, and have numerous applications. The asymptotic in the interior of the support of μ, is quite different from that at the edges, or in the exterior. In this paper, we focus on asymptotics at the edges. The best known such asymptotic is the Mehler-Heine formula for classical Jacobi polynomials { P (α,β) n } , which are orthogonal with respect to the Jacobi weight (1.1) w (x) = (1− x) (1 + x) , x ∈ (−1, 1) , Date : January 2, 2018. 1

Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality