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

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.