Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the $D_{5}$ Sakai surface
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| Format: | Preprint |
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2025
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| _version_ | 1866909889906868224 |
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| author | Zhu, Mengkun Chen, Siqi Zhang, Xuhao |
| author_facet | Zhu, Mengkun Chen, Siqi Zhang, Xuhao |
| contents | In this paper, we focus on the relationship between the d-P$\left(A_{3}^{(1)}/D_{5}^{(1)}\right)$ equations and a time-evolved Jacobi weight, $w(x)=x^α(1-x)^β\mathrm{e}^{-sx}$, $x\in[0,1]$, $α,β> -1$, $s>0$. From the perspective of Sakai's geometric theory of Painlevé equations, we derive that a recurrence relation closely related to the recurrence coefficients of monic polynomials orthogonal with $w(x)$ is equivalent to the standard d-P$\left(A_{3}^{(1)}/D_{5}^{(1)}\right)$ equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the $D_{5}$ Sakai surface Zhu, Mengkun Chen, Siqi Zhang, Xuhao Classical Analysis and ODEs In this paper, we focus on the relationship between the d-P$\left(A_{3}^{(1)}/D_{5}^{(1)}\right)$ equations and a time-evolved Jacobi weight, $w(x)=x^α(1-x)^β\mathrm{e}^{-sx}$, $x\in[0,1]$, $α,β> -1$, $s>0$. From the perspective of Sakai's geometric theory of Painlevé equations, we derive that a recurrence relation closely related to the recurrence coefficients of monic polynomials orthogonal with $w(x)$ is equivalent to the standard d-P$\left(A_{3}^{(1)}/D_{5}^{(1)}\right)$ equation. |
| title | Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the $D_{5}$ Sakai surface |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.04176 |