Fuchsian differential equations of order 3,...,6 with three singular points and an accessory parameter II, Equations of order 3

TL;DR

This paper investigates third-order Fuchsian differential equations with three singular points and an accessory parameter, identifying shift operators for specific subfamilies and analyzing their properties.

Contribution

It introduces new shift operators for certain codimension-2 subfamilies of third-order Fuchsian equations with accessory parameters.

Findings

No shift operators for generic exponents.

Shift operators exist for specific subfamilies with polynomial accessory parameters.

The Dotsenko-Fateev equation is among the identified cases.

Abstract

Fuchsian differential equations of order 3 with three singular points and with an accessory parameter are studied. When local exponents are generic, no shift operator is found, for codimension-1 subfamilies, neither. We found shift operators for several codimension-2 subfamilies of which accessory parameter is assigned as a cubic polynomial in the local exponents. The Dotsenko-Fateev equation is one of them.