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Strong asymptotics of orthogonal polynomials with respect to exponential weights. (English) Zbl 1026.42024

In this excellent paper the authors study asymptotics for orthogonal polynomials with respect to the weights

$w\left(x\right)dx={e}^{-Q\left(x\right)}dx$

on the real line, with $Q\left(x\right)$ an even polynomial of degree $2m$ with positive leading coefficients

The main results cover

a. asymptotics for the leading and recurrence coefficients (Theorem 2.1),

b. Plancherel-Rotach asymptotics on the whole complex plane (Theorem 2.2),

c. asymptotic location of the zeros (Theorem 2.3).

The deep results are derived through recently developed methods and a reformulation as a Riemann-Hilbert problem due to A. S. Fokas, A. R. Its and A. V. Kitaev [Commun. Math. Phys. 142, 313-344 (1991; Zbl 0742.35047); ibid. 147, 395-430 (1992; Zbl 0760.35051)].

The solution of this Riemann-Hilbert problem is then subjected to a series of transformations, leading to deep asymptotic results.

The technical and delicate operations are described in detail and give the reader a good insight in the different techniques needed. In view of the intricacies of the methods, the length of the paper is just about right.

##### MSC:
 42C05 General theory of orthogonal functions and polynomials 30E25 Boundary value problems, complex analysis 35Q15 Riemann-Hilbert problems