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 [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.

Keywords:

• Discrete Painlevé II
• orthogonal polynomials on the unit circle
• special solution
• upper bound

2020 Mathematics Subject Classifications:

• 33E17
• 39A13
• 42C05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by Fonds Wetenschappelijk Onderzoek (FWO) [grant number G0C9819N].