An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials P_n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P_n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n ^−1/3 for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.