Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions

Abstract

The structure of the extended affine Weyl symmetry group of higher Painlev\'e equations with periodicity $N$ varies depending on whether $N$ is even or odd. For even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ includes not only the conventional B\"acklund transformations $s_j, j=1,{\ldots},N$, and the group of automorphisms consisting of cycling permutations but also incorporates %encompasses an additional expansion of the group of automorphisms by embedding %featuring in this group the reflections on a periodic circle of $N$ points. This latter aspect is a novel feature revealed in this paper. The presence of reflection automorphisms is linked to the existence of degenerated solutions. Specifically, for $N=4$ we explicitly demonstrate how the reflection automorphisms around even points induce degeneracy in a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We provide closed-form expressions for both the solutions and their degenerated counterparts, given in terms of determinants of Kummer polynomials.