Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the $D_{5}$ Sakai surface
Mengkun Zhu, Siqi Chen, Xuhao Zhang
TL;DR
This work establishes a precise link between the recurrence coefficients of time-evolved Jacobi-weights and discrete Painlevé dynamics on the $D_{5}^{(1)}$ Sakai surface. By deploying Sakai's geometric framework, the authors derive a two-component recurrence for $x_n$ and $y_n$, with $x_n = \frac{1}{s}-\frac{1}{R_{n-1}(s)}$ and $y_n = -r_n(s)$, which is shown to be equivalent to the standard discrete Painlevé equation $d-P(A_{3}^{(1)}/D_{5}^{(1)})$ under a birational coordinate change. They provide an explicit coordinate transformation $x(f,g) = -\frac{ f (g+t) + n }{ t ( f g + n) }$, $y(f,g) = \frac{(f g + n)(g+t)}{t}$ with $s(t)=-t$, together with its inverse, and identify the underlying root-variable normalization $a_{0}=n+\beta$, $a_{1}=-n$, $a_{2}=n+\alpha$, $a_{3}=1-n-\alpha-\beta$. The proof combines singularity analysis, surface blowups, and Picard-lattice comparisons to match the time-evolved Jacobi recurrence with the standard $d-P(A_{3}^{(1)}/D_{5}^{(1)})$ dynamics, thereby linking orthogonal-polynomial recurrence theory to discrete Painlevé geometry and enabling coordinate-free interpretation of the coefficients.
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, $w(x)=x^α(1-x)^β\mathrm{e}^{-sx}$, $x\in[0,1]$, $α,β> -1$, $s>0$. 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.