zbMATH — the first resource for mathematics

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.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)