Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight

TL;DR

This paper analyzes the asymptotic behavior of recurrence coefficients for polynomials orthogonal with a logarithmic weight, confirming a conjecture using advanced Riemann-Hilbert techniques despite the singularity challenge.

Contribution

It computes the asymptotics of orthogonal polynomial recurrence coefficients with a logarithmic weight and verifies Magnus's conjecture using novel Riemann-Hilbert methods.

Findings

Confirmed Magnus's conjecture for recurrence coefficients.

Derived asymptotic formulas for orthogonal polynomials with logarithmic weights.

Developed non-standard Riemann-Hilbert analysis techniques.

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 $x=1$.