Unique special solution for discrete Painlevé II

We show that the discrete Painlevé II equation with starting value $a_{-1}=-1$ has a unique solution for which $-1<a_n<1$ for every $n\ge 0$. This solution corresponds to the Verblunsky coefficients of a family of orthogonal polynomials on the unit circle. This result was already proved for certain values of the parameter in the equation and recently a full proof was given by Duits and Holcomb. In the present paper we give a different proof that is based on an idea put forward by Tomas Lasic Latimer which uses orthogonal polynomials. We also give an upper bound for this special solution.