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

TL;DR

This paper presents a particular solution to a generalized Painlevé VI system using hypergeometric functions and explores its degeneration structure.

Contribution

It introduces a specific solution of a higher-order Painlevé system in terms of hypergeometric functions and analyzes its degeneration from confluence.

Findings

Solution expressed via hypergeometric function ${}_{n+1}F_n$

Degeneration structure derived from confluence of hypergeometric functions

Extension of Painlevé VI system with $A^{(1)}_{2n+1}$-symmetry

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