Shift relations and reducibility of some Fuchsian differential equations of order 2,...,6 with three singular points

Abstract

A Fuchsian differential equation of order six with nine free exponents as parameters and with three singular points is presented. This equation has various symmetries, which specify the accessory parameter as a polynomial of the local exponents. For some shifts of exponents, the shift operators are found, which lead to reducibility conditions of the equation. By several specializations of the parameters and successive factorizations of the equation, it produces several new equations and also known ones. For each equation, shift operators are studied, and when the equation is reducible, we observe the way of factorization and discuss the relation with the shift operators.