Weights with both absolutely continuous and discrete components:   Asymptotics 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 Pollazek Polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollazek 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. 
Xiao-Bo Wu, Yu Lin, Shuai-Xia Xu and Yu-Qiu Zhao
math.CV