Monotone Jacobi parameters and non-Szego weights

Abstract

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\equiv 1$, $b_n =-C n^{-\beta}$ ($0<\beta< \frac23)$, one has $d\mu(x)= w(x) dx$ on $(-2,2)$, and near $x=2$, $w(x)=e^{-2Q(x)}$ where \[ Q(x)=\beta^{-1} C^{\frac{1}{\beta}} \frac{\Gamma(\frac32)\Gamma(\frac{1}\beta}-\frac12)(2-x)^{\frac12 -\frac{1}{\beta}}}{\Gamma(\frac{1}{\beta}+1)}(1+O((2-x))) \]