Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach

Let Q(x) = q 2m x 2m + q 2m-1 x 2m-1 + ... be a polynomial of degree 2m with q 2m > 0, and let {π n (x)} n≥1 be the sequence of monic polynomials orthogonal with respect to the weight w(x) = e -Q(x) on R. Furthermore, let α n and β n denote the Mhaskar-Rakhmanov-Saff (MRS) numbers associated with Q(x). By using the Riemann-Hilbert approach, an asymptotic expansion is constructed for π n (c n z + d n ), which holds uniformly for all z bounded away from (-∞, -1), where c n = 1 2(β n - α n ) and d n = 1 2(β n + α n ).