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

TL;DR

This paper explores the connection between discrete Painlevé equations and the recurrence coefficients of orthogonal polynomials with a time-evolved Jacobi weight, using Sakai's geometric framework to establish their equivalence.

Contribution

It demonstrates that a recurrence relation for orthogonal polynomial coefficients is equivalent to a specific discrete Painlevé equation within Sakai's geometric classification.

Findings

Recurrence coefficients relate to discrete Painlevé equations

Established geometric interpretation of the recurrence relations

Connected orthogonal polynomial theory with Sakai's Painlevé 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, $w(x)=x^{\alpha}(1-x)^{\beta}\mathrm{e}^{-sx}$, $x\in[0,1]$, $\alpha,\beta > -1$, $s>0$. 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.