Uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation

Abstract

We study a class of sieved Pollaczek polynomials defined by a second-order difference equation (three-term recurrence relation). The measure of orthogonality is determined by using the Markov theorem and the Perron–Stieltjes inversion formula, and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points. Uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point α n and a critical point β n , involving respectively the Airy function Ai, and A = − 1 2 Ai + i 2 Bi . Darboux's method, the method of steepest descents, and various uniform asymptotic techniques such as cubic transformations are used to derive the results. Asymptotic formulas for the least zeros, the largest zeros, and the zeros on both sides of β n are also obtained. Several numerical examples are provided to compare the approximate zeros with the true values.