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

Siqi Chen, Mengkun Zhu

TL;DR

This work connects the recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^{\lambda} e^{-x^2+s x}$ to discrete Painlevé dynamics within Sakai’s geometric framework. By transforming the ladder-operator expressions of Filipuk et al., the authors obtain a recurrence for variables $x_n,y_n$ that is equivalent to the standard discrete Painlevé equation in the $d$-$P(A_{2}^{(1)}/E_{6}^{(1)})$ family via a detailed identification procedure. They perform a thorough singularity analysis, blow up the base points on $\mathbb{P}^1\times\mathbb{P}^1$, track the induced Picard-lattice action, compute the period map, and construct an explicit birational coordinate change between $(q,p,t)$ and $(x,y)$; this yields the root-variable relations $a_0=1-\lambda$, $a_1=-n$, $a_2=n+\lambda$, and confirms that the Laguerre-recurrence is governed by the canonical $d$-$P_{VI}$-type dynamics on the $E_{6}^{(1)}$ surface. The result integrates semiclassical Laguerre recurrences into Sakai’s classification and provides explicit transformations for practical computation of the coefficients. $

Abstract

In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^λ\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $λ>-1$, $s\in\mathbb{R}$, and Painlevé 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.

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

TL;DR

This work connects the recurrence coefficients of the semiclassical Laguerre weight to discrete Painlevé dynamics within Sakai’s geometric framework. By transforming the ladder-operator expressions of Filipuk et al., the authors obtain a recurrence for variables that is equivalent to the standard discrete Painlevé equation in the - family via a detailed identification procedure. They perform a thorough singularity analysis, blow up the base points on , track the induced Picard-lattice action, compute the period map, and construct an explicit birational coordinate change between and ; this yields the root-variable relations , , , and confirms that the Laguerre-recurrence is governed by the canonical --type dynamics on the surface. The result integrates semiclassical Laguerre recurrences into Sakai’s classification and provides explicit transformations for practical computation of the coefficients. $

Abstract

In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight , , , , and Painlevé 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 equations.

Paper Structure

This paper contains 14 sections, 8 theorems, 72 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Discrete Painlevé Equations in the d-$\text{P}\left(\mathrm{A_{2}^{(1)}/E_{6}^{(1)}}\right)$ family
  3. The Point Configuration
  4. The Extended Affine Weyl Symmetry Group
  5. Discrete Painlevé equations on the $E_{6}^{(1)}$ surface
  6. The Identification Procedure
  7. The Singularity Structure
  8. The Induced Mapping on $\operatorname{Pic}(\mathcal{X})$
  9. The Surface Type
  10. Initial Geometry Identification
  11. The Symmetry Roots and the Translations
  12. Final Geometry Identification
  13. The Period Map and the Identification of Parameters
  14. The Change of Coordinates

Key Result

Theorem 1

The recurrence eq:xyn-evol is equivalent to the standard discrete Painlevé equation eq:dPE6-KNY written in KajNouYam:2017:GAOPE. This equivalence is achieved via the following change of variables: The inverse change of variables is given by The relationship between the semiclassical Laguerre weight recurrence parameters and the root variables of discrete Painlevé equations is given by

Figures (7)

  • Figure 1: The standard surface root basis for the $E_6^{(1)}$ surface sub-lattice.
  • Figure 2: The standard symmetry root basis for the $A_2^{(1)}$ symmetry sub-lattice.
  • Figure 3: The model Sakai surface for the d-$\mathrm{P}\left(A_2^{(1)}/E_6^{(1)}\right)$ example.
  • Figure 4: The surface root basis for the semiclassical Laguerre weight recurrence.
  • Figure 5: The Sakai surface for the semiclassical Laguerre weight recurrence.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 5 more