Unique special solution for discrete Painlev\'e II

TL;DR

This paper proves the uniqueness of a special bounded solution to the discrete Painlevé II equation, linking it to orthogonal polynomials on the unit circle and providing a new proof approach and bounds.

Contribution

It offers a novel proof of the unique bounded solution for discrete Painlevé II using orthogonal polynomial techniques, extending previous results.

Findings

Established the uniqueness of the special solution with boundedness property.

Connected the solution to Verblunsky coefficients of orthogonal polynomials.

Provided an upper bound for this particular solution.

Abstract

We show that the discrete Painlev\'e II equation with starting value $a_{-1}=-1$ has a unique solution for which $-1 < a_n < 1$ for every $n \geq 0$. 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.