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Unique special solution for discrete Painlevé II

Walter Van Assche

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 $n \geq 0$. 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.

Unique special solution for discrete Painlevé II

Abstract

We show that the discrete Painlevé II equation with starting value has a unique solution for which for every . 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.
Paper Structure (4 sections, 4 theorems, 58 equations)

This paper contains 4 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials on the unit circle
  3. Proof of Theorem \ref{['thm1']}
  4. Bounds for the unique solution

Key Result

Theorem 1

The discrete Painlevé II equation with initial value $a_{-1}=-1$ has a unique solution for which $-1 < a_n < 1$ for all $n \geq 0$. This solution is obtained by taking $a_0 = I_1(t)/I_0(t)$, where $I_n(t)$ is the modified Bessel function of order $n$.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof