Recurrence coefficients for the time-evolved Jacobi weight and discrete Painlevé equations on the $D_5$ Sakai surface

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, $w(x)=x^{\hat{I}}\pm (1-x{)}^2{\mathrm{e}}^{-sx}$, $x\in [0,1]$, $\hat{I}\pm ,{\hat{I}}^2>-1$, $s>0$. 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.

9 pages