Rational solutions of higher order Painlev\'{e} systems I

This is the first paper of a series whose aim is to reach a complete classification and an explicit representation of rational solutions to the higher order generalizations of $\textrm{PIV}$ and $\textrm{PV}$, also known as the $A_N$-Painlev\'e or the Noumi-Yamada system. This paper focuses on the construction of rational solutions for the $A_{2n}$-Painlev\'e system. In this even case, we introduce a method to construct these rational solutions based on cyclic dressing chains of Schr\"{o}dinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and we characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto and Umemura polynomials, showing that they are particular cases of a larger family.