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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.

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