Extended symmetry of higher Painlev\'e equations of even periodicity and their rational solutions

TL;DR

This paper explores the extended symmetry structure of higher Painlevé equations with even periodicity, revealing novel reflection automorphisms that influence rational solutions and their degeneracies, with explicit solution formulas provided.

Contribution

It uncovers the presence of reflection automorphisms in the symmetry group for even N, a novel feature, and links these to solution degeneracies, providing explicit determinant formulas.

Findings

Reflection automorphisms exist for even N in the symmetry group.

Reflection automorphisms cause degeneracy in rational solutions.

Explicit solutions are expressed via determinants of Kummer polynomials.

Abstract

The structure of extended affine Weyl symmetry group of higher Painlev\'e equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional B\"acklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials.