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On asymptotic properties of certain orthogonal polynomials. (English) Zbl 0897.33007
Let $$V(\lambda)$$ be a real valued function on the whole real axis $$\mathbb{R}$$ and consider the polynomials $$P^{(n)}_l(\lambda)$$, $$l= 0,1,\dots$$, where $$n$$ is a large parameter and which are orthogonal on $$\mathbb{R}$$ with respect to the weight $$w_n(\lambda)= \exp[-nV(\lambda)]$$. This paper is concerned with the asymptotic properties of these polynomials and related quantities as $$n\to\infty$$. In particular, the asymptotic distribution of the zeros is considered. The techniques used and the results are motivated by studies on the eigenvalues statistics of random matrices.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
##### Keywords:
asymptotic distribution of zeros