Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight

Abstract

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x){\rm d}x = \log \frac{2k}{1-x}{\rm d}x$ on $(-1,1)$, $k > 1$, and verify a conjecture of A. Magnus for these coefficients. We use Riemann-Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann-Hilbert problem in a neighborhood of the logarithmic singularity at $x=1$.