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

Abstract

. We obtain a strong asymptotic formula for the leading coefficient α 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 − 2 nQ ( 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 α n ( n ) e − nw Q oscillates as n → ∞ and, in a typical case, fills an interval (here w Q is the equilibrium constant in the external field Q ).