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 [A double scaling limit for the d-PII equation with boundary conditions. arXiv:2304.02918 [math.CA]]. In the present paper we give a different proof that is based on an idea put forward by Tomas Lasic Latimer [Unique positive solutions to q-discrete equations associated with orthogonal polynomials, J. Difference Equ. Appl. 27 (2021), pp. 763–775.] which uses orthogonal polynomials. We also give an upper bound for this special solution.