A series solution of the general Heun equation in terms of incomplete Beta functions

TL;DR

This paper presents a series expansion of the general Heun equation using incomplete Beta functions, deriving closed-form solutions and explicit coefficients under specific parameter conditions.

Contribution

It introduces a novel series solution in terms of incomplete Beta functions and provides explicit formulas for coefficients and closed-form solutions for special parameter sets.

Findings

Series expansion in incomplete Beta functions for specific parameter cases

Closed-form solutions derived from series termination

Explicit coefficients obtained via recurrence relations

Abstract

We show that in the particular case when a characteristic exponent of the singularity at infinity is zero the solution of the general Heun equation can be expanded in terms of the incomplete Beta functions. By means of termination of the series, closed-form solutions are derived for two infinite sets of the involved parameters. These finite-sum solutions are written in terms of elementary functions that in general are quasi-polynomials. The coefficients of the expansion obey a three-term recurrence relation, which in some particular cases may become two-term. We discuss the case when the recurrence relation involves two non-successive terms and show that the coefficients of the expansion are then calculated explicitly and the general solution of the Heun equation is constructed as a combination of several hypergeometric functions with quasi-polynomial pre-factors.