Elliptic asymptotic behaviour of $q$-Painlevé transcendents
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915330592342016 |
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| author | Holroyd, Joshua |
| author_facet | Holroyd, Joshua |
| contents | The discrete Painlevé equations have mathematical properties closely related to those of the differential Painlevé equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlevé transcendents, analogous to the asymptotic theory of classical Painlevé transcendents. We focus on the $q$-difference second Painlevé equation in the asymptotic regime $|q-1|\ll1$, showing that generic leading-order behaviour is given in terms of elliptic functions and that the slow modulation in this behaviour is approximated in terms of complete elliptic integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05724 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic asymptotic behaviour of $q$-Painlevé transcendents Holroyd, Joshua Classical Analysis and ODEs Mathematical Physics The discrete Painlevé equations have mathematical properties closely related to those of the differential Painlevé equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlevé transcendents, analogous to the asymptotic theory of classical Painlevé transcendents. We focus on the $q$-difference second Painlevé equation in the asymptotic regime $|q-1|\ll1$, showing that generic leading-order behaviour is given in terms of elliptic functions and that the slow modulation in this behaviour is approximated in terms of complete elliptic integrals. |
| title | Elliptic asymptotic behaviour of $q$-Painlevé transcendents |
| topic | Classical Analysis and ODEs Mathematical Physics |
| url | https://arxiv.org/abs/2506.05724 |