Asymptotics of Derivatives of Orthogonal Polynomials on the Real Line

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L2 norm. 1. Results Let μ be a nite positive Borel measure on [−1, 1] and let {pn}n=0 denote the corresponding orthonormal polynomials, so that ∫ 1 −1 pnpm dμ = δmn. Asymptotics for derivatives of pn have been established under various hypotheses [1], [2], [9], [10], [13]. Many of these results deal with orthogonal polynomials on the unit circle. Recall that corresponding to μ, we may de ne ∗Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.