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

TL;DR

This paper introduces Fuchsian differential equations of orders 3 to 6 with three singular points and an accessory parameter, analyzing shift operators for order 6 to reveal symmetries and polynomial relations.

Contribution

It presents new classes of Fuchsian equations with specific singularity structures and explores shift operators to understand their symmetries and accessory parameters.

Findings

Shift operators for order 6 equations are studied.

Accessory parameter of order 6 is expressed as a cubic polynomial.

Equations exhibit several symmetries related to local exponents.

Abstract

Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with three singular points and one accessory parameter are presented. The shift operators for $H_6$ are studied. They lead to assign the accessory parameter of $H_6$ a cubic polynomial of local exponents so that the equation has several nice symmetries. The other equations will be studied in the forthcoming papers.