A higher order Painlev\'e system in two variables and extensions of the Appell hypergeometric functions $F_1$, $F_2$ and $F_3$

TL;DR

This paper introduces an extension of the Appell hypergeometric functions $F_2$, $F_3$, and $F_1$, derived from a higher order Painlevé system in two variables, revealing their equivalence at the PDE system level.

Contribution

It presents a novel extension of Appell hypergeometric functions based on a higher order Painlevé system, unifying different extensions through PDE analysis.

Findings

Extension of $F_2$ and $F_3$ derived from Painlevé system

Extension of $F_1$ introduced by Tsuda

Equivalence of these extensions at the PDE system level

Abstract

In this article we propose an extension of Appell hypergeometric function $F_2$ (or equivalently $F_3$). It is derived from a particular solution of a higher order Painlev\'e system in two variables. On the other hand, an extension of Appell's $F_1$ was introduced by Tsuda. We also show that those two extensions are equivalent at the level of systems of linear partial differential equations.