Orthonormal polynomials on the unit circle and spatially discrete PainlevéII equation
Chie Bing Wang; Chie Bing Wang; Department of Mathematics and Institute of Theoretical Dynamics University of California, Davis, CA 95616, USA
Журнал:
Journal of Physics A: Mathematical and General
Дата:
1999-10-15
Аннотация:
We consider the polynomials <sub>n</sub>(z) = <sub>n</sub>(z<sup>n</sup>+b<sub>n-1</sub>z<sup>n-1</sup>+...) orthonormal with respect to the weight exp(()<sup>1/2</sup>(z+1/z)) dz/2iz on the unit circle in the complex plane. The leading coefficient <sub>n</sub> 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 Painlevé II equation. The leading coefficient and second leading coefficient of <sub>n</sub>(z) can be expressed asymptotically in terms of the Painlevé II function.
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