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arXiv · 2511.04176

Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlev\'e equations on the $D_{5}$ Sakai surface

Mengkun Zhu·Siqi Chen·Xuhao Zhang
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Abstract

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, $w(x)=x^{\alpha}(1-x)^{\beta}\mathrm{e}^{-sx}$, $x\in[0,1]$, $\alpha,\beta > -1$, $s>0$. From the perspective of Sakai's geometric theory of Painlev\'e 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.

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Mengkun Zhu, Siqi Chen, Xuhao Zhang. 2025-11-06. Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlev\'e equations on the $D_{5}$ Sakai surface. https://arxiv.org/abs/2511.04176

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