Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation

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.