Narrow Results

Let {p_n(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e^−υ(x), where υ(x) = x^2m + … is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for p_n(x), as n→∞, which holds uniformly for 0 ≤ x ∞ O(n^½m). As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

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 → ∞.

A turning-point theory is developed for the second-order difference equation