Semiclassical limit of a non-polynomial $q$-Askey scheme
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910512691806208 |
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| author | Lenells, Jonatan Roussillon, Julien |
| author_facet | Lenells, Jonatan Roussillon, Julien |
| contents | We prove a semiclassical asymptotic formula for the two elements $\mathcal M$ and $\mathcal Q$ lying at the bottom of the recently constructed non-polynomial hyperbolic $q$-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlevé I and $\textrm{III}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlevé equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlevé equations in the semiclassical limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical limit of a non-polynomial $q$-Askey scheme Lenells, Jonatan Roussillon, Julien Classical Analysis and ODEs We prove a semiclassical asymptotic formula for the two elements $\mathcal M$ and $\mathcal Q$ lying at the bottom of the recently constructed non-polynomial hyperbolic $q$-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlevé I and $\textrm{III}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlevé equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlevé equations in the semiclassical limit. |
| title | Semiclassical limit of a non-polynomial $q$-Askey scheme |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.03464 |