A study of a Fuchsian system of rank 8 in 3 variables and the ordinary differential equations as its restrictions

TL;DR

This paper investigates a complex Fuchsian system of rank 8 in three variables, exploring its singularities, reductions to ordinary differential equations, and connections to hypergeometric functions and the Dotsenko-Fateev equation.

Contribution

It introduces a specific Fuchsian system of rank 8, analyzes its restrictions, and links it to hypergeometric and Dotsenko-Fateev equations through middle convolution techniques.

Findings

Derived explicit local solutions with coefficients as sums of Gamma functions.

Connected the system's reductions to well-known hypergeometric equations.

Expressed solutions as special values of the $_4F_3$ hypergeometric series.

Abstract

A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equation, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions to these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series $_4F_3$ at 1. Keywords: Fuchsian differential equation, hypergeometric differential equation, middle convolution, Dotsenko-Fateev equation, recurrence formula, series solution