dorsal/arxiv
View SchemaOrthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation
| Authors | Chie Bing Wang |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9902011 |
| URL | https://arxiv.org/abs/solv-int/9902011 |
| DOI | 10.1088/0305-4470/32/41/312 |
Abstract
We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z)) dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient $\kappa_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlev\'e II equation. The leading coefficient and second leading coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the Painlev\'e II function.
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"abstract": "We consider the polynomials $\\phi_n(z)= \\kappa_n (z^n+ b_{n-1} z^{n-1}+\n\u003e...)$ orthonormal with respect to the weight $\\exp(\\sqrt{\\lambda} (z+ 1/z))\ndz/2 \\pi i z$ on the unit circle in the complex plane. The leading coefficient\n$\\kappa_n$ is found to satisfy a difference-differential (spatially discrete)\nequation which is further proved to approach a third order differential\nequation by double scaling. The third order differential equation is equivalent\nto the Painlev\\\u0027e II equation. The leading coefficient and second leading\ncoefficient of $\\phi_n(z)$ can be expressed asymptotically in terms of the\nPainlev\\\u0027e II function.",
"arxiv_id": "solv-int/9902011",
"authors": [
"Chie Bing Wang"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1088/0305-4470/32/41/312",
"title": "Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\\\u0027e II Equation",
"url": "https://arxiv.org/abs/solv-int/9902011"
},
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