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Strong asymptotics of polynomials orthogonal with respect to Freud weights. (English) Zbl 0944.42014
In this important and interesting paper, the authors again illustrate the power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Moreover, there is a new feature: they show how the technique may be applied, with suitable modifications, even in the presence of singularities. Let \(\alpha>0\) and \[ w(x):= \exp(-|x|^\alpha),\quad \alpha>0,\;x\in\mathbb{R}. \] Then we may define orthonormal polynomials \[ p_n(x)= \gamma_n x^n+\cdots,\quad \gamma_n> 0, \] satisfying \[ \int p_n p_m w= \delta_{mn},\quad m,n\geq 0. \] The authors obtain very precise (Plancherel-Rotach) asymptotics for the orthonormal polynomials in all regions of the plane, asymptotic relations for \(\gamma_n\) with error terms, the zeros of \(p_n\), the spacing between successive zeros, and so on.
There are two remarkable features in this particular work: firstly, they can handle not just \(\alpha\) an even positive integer, as was the case in their earlier papers, but also non-integer \(\alpha\); secondly, the case \(\alpha\leq 1\) is particularly difficult and defied efforts at asymptotics using other methods, such as Bernstein-Szegö techniques. So this is one case, where Riemann-Hilbert techniques not only yield more precise results, but also work for weights where other methods failed to give any asymptotics at all.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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