A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations

TL;DR

This paper reviews series solutions of the general and single-confluent Heun equations, focusing on hypergeometric function expansions, conditions for finite sums, recurrence relations, and solutions expressed via generalized hypergeometric functions.

Contribution

It provides a comprehensive analysis of hypergeometric solutions to Heun equations, including explicit parameter determination and conditions for finite and infinite series expansions.

Findings

Conditions for finite sum solutions are identified.

Explicit roots of polynomial equations for hypergeometric parameters are derived.

Many solutions can be expressed through a single generalized hypergeometric function.

Abstract

We review the series solutions of the general and single-confluent Heun equations in terms of powers, ordinary-hypergeometric and confluent-hypergeometric functions. The conditions under which the expansions reduce to finite sums as well as the cases when the coefficients of power-series expansions obey two-term recurrence relations are discussed. Infinitely many cases for which a solution of a general or single-confluent Heun equation is written through a single generalized hypergeometric function are indicated. It is shown that the parameters of this generalized hypergeometric function are the roots of a certain polynomial equation, which we explicitly present for any given order.