Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|^2αe^−(x4+tx2), x ∈ ℝ, where α is a constant larger than −½ and t is any real number. They consider this problem in three separate cases: (i) c > −2, (ii) c = −2, and (iii) c < −2, where c := tN^−½ is a constant, N = n + α and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μ_t is a single interval, whereas in the third case the support of μ_t consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). © The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2018.