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 $n\ge 0$. 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.

arXiv comment10 pages

References in corpus (4)

• Geometric Aspects of Painlevé Equations
• Unique positive solution for an alternative discrete Painlevé I equation
• A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions
• Unique positive solutions to $q$-discrete equations associated with orthogonal polynomials

Cited by in corpus (1)

• Special solutions of a discrete Painlevé equation for quantum minimal surfaces