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

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.