  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.
