Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

Abstract

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 −1/2 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 ^−1/2 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 μt is a single interval, whereas in the third case the support of μt 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).

Keywords

Orthogonal polynomials / Globally uniform asymptotics / Riemann-Hilbert problems / The second Painlevé transcendent / Theta function

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