On the discrete Painlev\'e equivalence problem, non-conjugate translations and nodal curves

TL;DR

This paper explores the classification of discrete Painlevé equations derived from orthogonal polynomial systems, highlighting the importance of considering surface types, symmetries, and non-conjugate dynamics for accurate correspondence.

Contribution

It demonstrates the necessity of a refined approach to the discrete Painlevé equivalence problem, incorporating group elements, surface types, and nodal curves for better classification.

Findings

Identified examples with common surface type $D_5^{(1)}$ but inequivalent dynamics.

Computed symmetry groups, including $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}.

Highlighted the role of nodal curves and non-conjugate elements in classifying discrete Painlevé equations.

Abstract

We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlev\'e equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type $D_5^{(1)}$ but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of $\widehat{W}(A_3^{(1)})$. Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}$. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlev\'e equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves.