An asymptotic formula for polynomials orthonormal with respect to a varying weight

Abstract:

We obtain a strong asymptotic formula for the leading coefficient $\alpha _{n}(n)$ of a degree $n$ polynomial $q_{n}(z;n)$ orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on $n$ as $e^{-2nQ(x)}$, where $Q(x)$ is a polynomial and corresponds to the “hard-edge case”. The formula in Theorem 1 is quite similar to Widom’s classical formula for a weight independent of $n$. In some sense, Widom’s formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that $\alpha _{n}(n)e^{[b]{-nw^{}_Q}}$ oscillates as $n\to \infty$ and, in a typical case, fills an interval (here $w_Q$ is the equilibrium constant in the external field $Q$).