*(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

Then we may define orthonormal polynomials

satisfying

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 $\alpha \le 1$ is particularly difficult and defied efforts at asymptotics using other methods, such as Bernstein-SzegĂ¶ 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 | General theory of orthogonal functions and polynomials |