Hypergeometric expansions of the general Heun function governed by two-term recurrence relations

TL;DR

This paper identifies specific parameter choices in the general Heun equation where the three-term recurrence relations simplify to two-term relations, allowing explicit solutions in terms of gamma functions.

Contribution

It demonstrates the existence of infinitely many parameter sets that reduce the recurrence relations to two terms, providing explicit formulas for the coefficients.

Findings

Infinite parameter sets yield two-term recurrence relations.

Explicit gamma function expressions for coefficients.

Simplification facilitates solving the general Heun equation.

Abstract

We show that there exist infinitely many particular choices of parameters for which the three-term recurrence relations governing the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions become two-term. In these cases the coefficients are explicitly written in terms of the gamma functions.