On the Form of Solutions of Fuchsian differential Equations with n regular singular Points

TL;DR

This paper characterizes the coefficients of power series solutions to Fuchsian differential equations with n regular singular points by solving recurrence relations, linking solutions to known special functions in physics.

Contribution

It provides a general method for determining power series coefficients of Fuchsian equations and explores special cases related to well-known special functions.

Findings

Coefficients are determined by solving n-term recurrence relations.

Special cases correspond to classical special functions.

Method applies to degenerated confluent forms of Fuchsian equations.

Abstract

The form of the coefficients of power series expressions corresponding to solutions of Fuchsian differential equations (or their associated degenerated confluent forms) with n regular singular points is determined by solving the corresponding n-term recurrence relations in full generality. Some important special cases are discussed in which the solutions coincide with special functions of mathematical physics.