Recurrence coefficients for orthogonal polynomials with a logarithmic weight function

Percy Deift, Mateusz Piorkowski

Published 2023-07-18Version 1

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \big(\frac{2}{1-x}\big) dx$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by A. Magnus and extends earlier results by T. O. 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 $x = +1$.