On the Asymptotics of the Meixner–Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner-Pollaczek polynomials M_n (n_α; δ, η) as n → ∞, which holds uniformly for −M ≤ α ≤ α M, where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α_n,s denotes the sth zero of Mn (n_α; δ, η), counted from the right, and if , denotes its sth zero counted from the left, then for each fxed s, three-term asymptotic approximations are obtained for both α_n,s and , as n → ∞.