Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation

Details Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation

1999-02-17

arxiv.org/abs/solv-int/9902011v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

16 pages

Scientific paper

10.1088/0305-4470/32/41/312

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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