On Certain Solutions for Confluent and Double-Confluent Heun Equations

TL;DR

This paper explores solutions to confluent and double-confluent Heun equations, reviewing existing series solutions, introducing new expansions, and establishing relations with solvable problems.

Contribution

It introduces an additional series expansion for confluent Heun solutions and derives conditions for linear combinations, expanding the solution space.

Findings

New series expansion for confluent Heun equations

Conditions for linear combination of solutions

Relations between Heun equations and solvable problems

Abstract

This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present problems which are ruled by each of these four equations and establish relations among Heun equations and quasi-exactly solvable problems.