Plancherel-Rotach type asymptotics of the sieved Pollaczek polynomials via the Riemann-Hilbert approach

摘要

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollaczek polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollaczek polynomials are obtained in the whole complex plane. The Riemann-Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band. (C) 2016 Elsevier Inc. All rights reserved.