Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters

Abstract

In this Note, we study the uniform asymptotics of the Meixner–Pollaczek polynomials P n ( λ n ) ( z ; ϕ ) with varying parameter λ n = ( n + 1 2 ) A as n → ∞ , where A > 0 is a constant. Uniform asymptotic expansions in terms of parabolic cylinder functions and elementary functions are obtained for z in two overlapping regions which together cover the whole complex plane.