Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914636357435392 |
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| author | Deift, Percy Piorkowski, Mateusz |
| author_facet | Deift, Percy Piorkowski, Mateusz |
| contents | 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 $x = +1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_09277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function Deift, Percy Piorkowski, Mateusz Classical Analysis and ODEs 42C05, 34M50, 45E05, 45M05 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 $x = +1$. |
| title | Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function |
| topic | Classical Analysis and ODEs 42C05, 34M50, 45E05, 45M05 |
| url | https://arxiv.org/abs/2307.09277 |