Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations

TL;DR

This paper investigates the recurrence coefficients of semiclassical Laguerre weights using Sakai's geometric framework, linking them to Painlevé equations and a specific class of discrete Painlevé equations, thereby revealing deep structural connections.

Contribution

It introduces a new transformation for ladder operator expressions and establishes a novel correspondence between recurrence relations and d-P(A2^(1)/E6^(1)) equations.

Findings

Derived a recurrence relation for Laguerre weight coefficients.

Established a link between recurrence relations and discrete Painlevé equations.

Enhanced understanding of the geometric structure underlying semiclassical orthogonal polynomials.

Abstract

In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^{\lambda}\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $\lambda>-1$, $s\in\mathbb{R}$, and Painlev\'e equations. Specifically, we introduce a new transformation for the expressions obtained by Filipuk et al. in their analysis of ladder operators for semiclassical Laguerre polynomials, thereby deriving a recurrence relation. Subsequently, we establish a correspondence between this recurrence relation and a class of d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations.