Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions

TL;DR

This paper develops new series expansions for solutions of the confluent Heun equation using incomplete Beta and Appell hypergeometric functions, analyzing recurrence relations and conditions for finite-sum solutions.

Contribution

It introduces novel expansions of the confluent Heun solutions in terms of special functions and examines their recurrence relations and finite-sum conditions.

Findings

Derived multiple series expansions with specific recurrence relations.

Identified conditions for finite-sum (closed-form) solutions.

Analyzed the structure of recurrence relations for these expansions.

Abstract

We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.