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In this paper, we study the uniform asymptotics of the Meixner-Pollaczek polynomials inline image with varying parameter inline image as inline image, where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann-Hilbert problems first introduced by Deift and Zhou [1].