Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight

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.