Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities

Ishkhanyan Ishkhanyan,Cesarano Cesarano

doi:10.3390/axioms8030102

Ishkhanyan Ishkhanyan, Cesarano Cesarano

Open Access

https://doi.org/10.3390/axioms8030102

Copy DOI

Abstract

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

Highlights

During the last decades, the five Heun equations and their generalizations became a subject of intensive investigations in the context of numerous advanced problems of contemporary fundamental and applied research
A computer algebra solver to find solutions having rational function arguments has been presented in [22]
The results we have reported here, that is the closed-form explicit solutions for equations having five regular singular points, are in line with similar results derived for the Heun-type equations [11,12]

Summary

Introduction

The five Heun equations and their generalizations became a subject of intensive investigations in the context of numerous advanced problems of contemporary fundamental and applied research. The general Heun equation, from which the other equations originate, is the most general Fuchsian second-order linear differential equation having four regular singular points [3,4]. Which presents the most general Fuchsian equation having five regular singular points [3,4]. Equation (1) becomes the general Heun equation with the third finite singularity being located at z = a1 and z = a, respectively For this reason, in this case, one may conventionally think of the parameters q and q1 as accessory parameters associated with the singularities a and a1 , respectively. A main result we report in the present paper is that there exist infinitely many particular choices of parameters for which Equation (1) having five irreducible regular singularities admits solutions in terms of a single generalized-hypergeometric function.

The Approach

Discussion