Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight

Let {p _n (x)}_{n\ge 0}$ be the set of orthonormal polynomials with respect to the exponential weight w(x)=e ^-v(x) , where v(x)=x ^2m + ... is a monic polynomial of degree 2m with $m \ge 2$ and is even. An asymptotic approximation is obtained for p _n (x), as $n \to \infty$, which holds uniformly for $0 \le x \le O(n^{1/2m})$. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.