Unique special solution for discrete Painlevé II

Abstract

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 ≥ 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 [3]. In the present paper we give a different proof that is based on an idea put forward by Tomas Lasic Latimer [6] which uses orthogonal polynomials. We also give an upper bound for this special solution.