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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.08343 |
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| _version_ | 1866911782575013888 |
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| author | Gibbons, John Stokes, Alexander Veselov, Alexander P. |
| author_facet | Gibbons, John Stokes, Alexander Veselov, Alexander P. |
| contents | We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes Gibbons, John Stokes, Alexander Veselov, Alexander P. Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry Classical Analysis and ODEs We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas. |
| title | Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2401.08343 |