A Particular Solution of a Painlev\'e System in Terms of the Hypergeometric Function ${}_{n+1}F_n$

Abstract

In a recent work, we proposed the coupled Painlev\'e VI system with $A^{(1)}_{2n+1}$-symmetry, which is a higher order generalization of the sixth Painlev\'e equation ($P_{\rm VI}$). In this article, we present its particular solution expressed in terms of the hypergeometric function ${}_{n+1}F_n$. We also discuss a degeneration structure of the Painlev\'e system derived from the confluence of ${}_{n+1}F_n$.