Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

TL;DR

This paper derives an asymptotic formula for the recurrence coefficients of orthogonal polynomials with a logarithmic weight, confirming a conjecture and extending previous results using the Riemann-Hilbert method.

Contribution

It provides the first asymptotic analysis for orthogonal polynomials with a logarithmic weight measure, addressing the challenge of a singularity at the endpoint.

Findings

Confirmed a special case of Magnus's conjecture.

Extended previous asymptotic results to a logarithmic weight.

Developed a Riemann-Hilbert approach for singular weights.

Abstract

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.