Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift; Mateusz Piorkowski

doi:10.3842/SIGMA.2024.004

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift, Mateusz Piorkowski

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure (formula presented). 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.

Original language	English (US)
Article number	004
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	20
DOIs	https://doi.org/10.3842/SIGMA.2024.004
State	Published - 2024

Keywords

orthogonal polynomials
recurrence coefficientsteepest descent method
Riemann–Hilbert problems

ASJC Scopus subject areas

Analysis
Mathematical Physics
Geometry and Topology

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10.3842/SIGMA.2024.004

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abstract = "We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure (formula presented). 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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author = "Percy Deift and Mateusz Piorkowski",

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AU - Deift, Percy

AU - Piorkowski, Mateusz

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AB - We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure (formula presented). 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.

KW - orthogonal polynomials

KW - recurrence coefficientsteepest descent method

KW - Riemann–Hilbert problems

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ER -

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Abstract

Keywords

ASJC Scopus subject areas

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