Orthonormal polynomials on the unit circle and spatially discrete Painleve 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(root lambda(z + 1/z)) dz/2 pi iz 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 Painleve II equation. The leading coefficient and second leading coefficient of phi(n) (z) can be expressed asymptotically in terms of the Painleve II function.