Generalized hypergeometric solutions of the Heun equation

TL;DR

This paper introduces a new class of solutions to the Heun equation expressed through generalized hypergeometric functions, under specific parameter restrictions involving characteristic exponents and accessory parameters.

Contribution

It provides infinitely many explicit solutions to the Heun equation using generalized hypergeometric functions with particular parameter constraints.

Findings

Solutions expressed in terms of generalized hypergeometric functions

Parameter restrictions include integer characteristic exponents and polynomial conditions on the accessory parameter

Enables new analytical approaches to solving the Heun equation

Abstract

We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the singularities should be a non-zero integer, and the accessory parameter must satisfy a polynomial equation.