Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients
Assume that the orthogonal polynomials \(p_ n(x)\), \(n=0,1,\dots\), are defined by the recursive equation \[ a_{n+1} p_{n+1}(x)+b_ n p_ n(x)+a_ n p_{n-1}(x)=xp_ n(x),\quad n=1,2,\dots, \] with \(p_{- 1}\equiv 0\) and \(p_ 1\equiv 1\) and the \(a_ n\)'s and \(b_ n\)'s are assumed to be regularly and slowly varying at infinity, i.e., there exists an increasing positive sequence \(\{\lambda_ n,\;n=0,1,\dots\}\) such that \[ \lim_{n\to\infty}{a_ n\over\lambda_ n}:= a>0,\quad\lim_{n\to\infty} {b_ n\over\lambda_ n}:= b\in\mathbb{R}, \] with \[ \lim_{n\to\infty} n\left({\lambda_{n+1}\over\lambda_ n}- 1\right):=\alpha\geq 0. \] Further it is assumed that \(a_{n+1}>0\), \(b_ n\in\mathbb{R}\), \(n=0,1,\dots\), and \[ \lim_{n\to\infty} n{a_{n+1}-a_ n\over\lambda_ n}=a\alpha,\quad\lim_{n\to\infty}n{b_{n+1}-b_ n\over \lambda_ n}=b\alpha. \] Polynomials orthogonal with respect to Freud and Erdős type weights belong to this class and have been subject of intense investigations. In the paper strong asymptotics of the contracted orthogonal polynomials \(p_ n(\lambda_ n y)\) have been investigated away from the interval of oscillation.
scientific article; zbMATH DE number 222122