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In this paper we study the asymptotic behavior of a family of polynomialswhich are orthogonal with respect to an exponential weight on certain contoursof the complex plane. The zeros of these polynomials are the nodes for complexGaussian quadrature of an oscillatory integral on the real axis with a highorder stationary point, and their limit distribution is also analyzed. We showthat the zeros accumulate along a contour in the complex plane that has theS-property in an external field. In addition, the strong asymptotics of theorthogonal polynomials is obtained by applying the nonlinear Deift--Zhousteepest descent method to the corresponding Riemann--Hilbert problem.

Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature