Unique special solution for discrete Painlevé II
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917568352092160 |
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| author | Van Assche, Walter |
| author_facet | Van Assche, Walter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07011 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unique special solution for discrete Painlevé II Van Assche, Walter Classical Analysis and ODEs 33E17, 39A13, 42C05, 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. |
| title | Unique special solution for discrete Painlevé II |
| topic | Classical Analysis and ODEs 33E17, 39A13, 42C05, |
| url | https://arxiv.org/abs/2308.07011 |