Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P n ( cos θ ; a , b ) . One is for θ ∈ ( 0 , δ / n ] , 0 < δ < a + b , in terms of elementary functions and in descending powers of n . The other is for θ ∈ [ δ / n , π / 2 ] , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P n ( cos θ ) in θ ∈ ( 0 , π / 2 ] .