On a fundamental system of solutions of a certain hypergeometric equation

TL;DR

This paper investigates the fundamental solutions of a class of hypergeometric equations, providing integral representations and exploring their connections to isomonodromic deformations and Painlevé equations.

Contribution

It introduces a fundamental system of solutions with specific local behavior for hypergeometric functions, linking them to isomonodromic deformation theory.

Findings

Constructed Euler-type integral representations for solutions

Established relations between hypergeometric systems and Painlevé equations

Analyzed local behavior of solutions in the hypergeometric context

Abstract

We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gau\ss's ${}_2 F_{1}$, Thomae's ${}_L F_{L-1}$ and Appell-Lauricella's $F_D$. In particular, we present a fundamental system of solutions with a characteristic local behavior by means of Euler-type integral representations. We also discuss how they are related to the theory of isomonodromic deformations or Painlev\'e equations.