Strong asymptotics of orthogonal polynomials with respect to exponential weights

We consider asymptotics of orthogonal polynomials with respect to weights w (x)dx = e(-Q(x))dx on the real line, where Q(x) = Sigma(k=0)(2m) q(k)x(k), q(2m) > 0, denotes a polynomial of even order with positive leading coeffic ient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further d eveloped in [17, 19] in order to obtain uniform Plancherel-Rotach-type asym ptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the or thogonal polynomials. (C) 1999 John Wiley & Sons, Inc.