Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function - Deift, Percy; Piorkowski, Mateusz (Symmetry, Integrability and Geometry: Methods and Applications, 2024)
We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure log(2/(1−))d 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 = +1.