Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

We consider the uniform asymptotics of polynomials orthogonal on [ 0,8) with respect to the exponential weight w(x) = x(alpha)e- (Q( x)), where alpha - 1 and Q( x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou [ Ann. Math. 137 ( 1993), 295-368].