On the asymptotics of the Meixner-Pollaczek polynomials and their zeros

An infinite asymptotic expansion is derived for the Meixner-Pollaczek polynomials M_n (nα; δ, η) as n → ∞, which holds uniformly for -M ≤ 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 M_n(nα; δ, η), counted from the right, and if α̃_n,s denotes its sth zero counted from the left, then for each fixed s, three-term asymptotic approximations are obtained for both α_n,s and α̃_n,s as n → ∞.