Incomplete beta-function expansions of the solutions to the confluent Heun equation

TL;DR

This paper develops new expansions of the confluent Heun equation solutions using incomplete Beta functions, including a novel type involving function combinations, and discusses conditions for obtaining elementary closed-form solutions.

Contribution

It introduces a new expansion type with Beta functions and provides criteria for when these expansions terminate into elementary functions.

Findings

New Beta-function expansions for confluent Heun solutions

Conditions for series termination into elementary functions

Introduction of a novel expansion involving Beta function combinations

Abstract

Several expansions of the solutions to the confluent Heun equation in terms of incomplete Beta functions are constructed. A new type of expansion involving certain combinations of the incomplete Beta functions as expansion functions is introduced. The necessary and sufficient conditions when the derived expansions are terminated, thus generating closed-form solutions, are discussed. It is shown that termination of a Beta-function series solution always leads to a solution that is necessarily an elementary function.