Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

T A Ishkhanyan,A M Ishkhanyan,T A Shahverdyan

doi:10.1155/2018/4263678

Abstract

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

Highlights

The general Heun equation [1,2,3], which is the most general second-order linear ordinary differential equation having four regular singular points, is currently widely encountered in physics and mathematics research
We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions
Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function

Summary

Introduction

The general Heun equation [1,2,3], which is the most general second-order linear ordinary differential equation having four regular singular points, is currently widely encountered in physics and mathematics research (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and references therein). We show that there exist some particular choices of the involved parameters for which the recurrence relations governing the power-series expansions become two-term In these cases the solution of the general Heun equation can be written either as a linear combination of a finite number of the Gauss hypergeometric functions or in terms of a single generalized hypergeometric function. As far as the expansions of the general Heun equation in terms of the hypergeometric functions are concerned, in the early papers by Svartholm [19], Erdelyi [20, 21], and Schmidt [22], the intuitive intention was to apply hypergeometric functions with parameters so chosen as to match the Heun equation as closely as possible For this reason they used functions of the form 2F1(λ + n, μ − n; γ; z), which have matching behavior in two singular points, z = 0 and 1. The general conclusion is that in any such case the power-series expansion of the Heun function is governed by a two-term recurrence relation (obviously, this is the relation obeyed by the corresponding power-series for pFq)

Hypergeometric Expansions

Finite-Sum Hypergeometric Solutions

Discussion