Unique special solution for discrete Painlevé II

We show that the discrete Painlevé II equation with starting value $ a_{-1}=-1 $ a − 1 = − 1 has a unique solution for which $ -1 { \lt } a_n { \lt } 1 $ − 1 < a n < 1 for every $ n \geq ~0 $ n ≥   0 . This solution corresponds to the Verblunsky coefficients of a family of orthogonal polynomia...

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Bibliographic Details
Published in:Journal of difference equations and applications 2024-04, Vol.30 (4), p.465-474
Main Author: Van Assche, Walter
Format: Article
Language:English
Subjects:
Boundary conditions
Discrete Painlevé II
orthogonal polynomials on the unit circle
Polynomials
special solution
Uniqueness
upper bound
Upper bounds
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Summary:We show that the discrete Painlevé II equation with starting value $ a_{-1}=-1 $ a − 1 = − 1 has a unique solution for which $ -1 { \lt } a_n { \lt } 1 $ − 1 < a n < 1 for every $ n \geq ~0 $ 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.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2023.2294919