Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 004, 48 pages      arXiv:2307.09277      https://doi.org/10.3842/SIGMA.2024.004
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift a and Mateusz Piorkowski b
a) Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Str., New York, NY 10012, USA
b) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

Received July 19, 2023, in final form January 01, 2024; Published online January 10, 2024

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 $x = +1$.

Key words: orthogonal polynomials; Riemann-Hilbert problems; recurrence coefficients; steepest descent method.

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