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 $w(x)\mathrm{d}x=\log \frac{2k}{1-x}\mathrm{d}x$ 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 $x=1$.

References in corpus (1)

• Large Deviations and Sum Rules for Spectral Theory - A Pedagogical Approach

Cited by in corpus (3)

• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics
• Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function
• Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials