Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation
Abstract
We consider the polynomials
ϕ
n
(z)=
κ
n
(
z
n
+
b
n−1
z
n−1
+>...)
orthonormal with respect to the weight
exp(
λ
−
−
√
(z+1/z))dz/2πiz
on the unit circle in the complex plane. The leading coefficient
κ
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 Painlevé II equation. The leading coefficient and second leading coefficient of
ϕ
n
(z)
can be expressed asymptotically in terms of the Painlevé II function.