  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 ${}_{n+1}F_n$.
