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arXiv · 2307.09277

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift·Mateusz Piorkowski
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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 $x = +1$.

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Percy Deift, Mateusz Piorkowski. 2023-07-18. Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function. https://doi.org/10.3842/sigma.2024.004

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