Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Abstract

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \left(\frac{2}{1-x}\right)\mathrm{d}x$ 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 $x=+1$.

References in corpus (1)

• Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight