New solutions of Heun general equation

TL;DR

This paper identifies special cases where derivatives of Heun equation solutions relate to other Heun solutions, and constructs series solutions using hypergeometric functions to address singular behaviors.

Contribution

It introduces new cases linking derivatives of Heun solutions to other Heun equations and develops series solutions with proper singular behavior using hypergeometric functions.

Findings

Derived conditions where derivatives relate to other Heun solutions

Constructed series solutions with correct singular behavior

Utilized hypergeometric functions for solution expansion

Abstract

We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behavior at only one of the singular points of the equation; the sum, however, has correct behavior.