Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift ; Mateusz Piorkowski

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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 $x = +1$.

Détail

Keywords: Riemann–Hilbert problems, recurrence coefficients, steepest descent method.
Mots-clés : orthogonal polynomials

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