# Expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions

## TL;DR

This paper develops new series expansions of the general Heun equation solutions using incomplete Beta functions and Appell hypergeometric functions, applicable for arbitrary parameters and including conditions for finite-sum solutions.

## Contribution

It introduces novel expansions of the Heun equation solutions in terms of special functions with recurrence relations and finite-sum conditions, broadening analytical solution methods.

## Key findings

- Expansions in terms of incomplete Beta functions and Appell hypergeometric functions.
- Recurrence relations for expansion coefficients involve four to six terms.
- Conditions for finite-sum solutions are identified.

## Abstract

Applying the approach based on the equation for the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions. Several expansions in terms of the Appell generalized hypergeometric functions of two variables of the fist kind are also presented. The constructed expansions are applicable for arbitrary sets of the involved parameters. The coefficients of the expansions obey four-, five- or six-term recurrence relations. However, there exist several sets of the parameters for which the recurrence relations involve fewer terms, not necessarily successive. The conditions for deriving finite-sum solutions via termination of the series are discussed.

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Source: https://tomesphere.com/paper/1703.08177