Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions

T A Ishkhanyan,A M Ishkhanyan

doi:10.1063/1.4893997

Abstract

We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions in general obey three-term recurrence relations defining double-sided infinite series; however, four-term and two-term relations are also possible in particular cases. The conditions for left- and/or right-side termination of the derived series are discussed.

Highlights

Expansions of the solutions of the confluent Heun equation[1,2,3] in terms of mathematical functions other than powers have been discussed by many authors
We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions
Using different recurrence relations obeyed by the Kummer confluent hypergeometric functions, we have constructed several confluent hypergeometric expansions of the solutions of the confluent Heun equation

Summary

INTRODUCTION

Expansions of the solutions of the confluent Heun equation[1,2,3] in terms of mathematical functions other than powers have been discussed by many authors (see, e.g., Refs. 3–11). Which slightly differs from that adopted in Ref. 3 since the parameters ε and α are here assumed to be independent This is a useful convention for practical applications since in this form the equation includes the Whittaker-Ince limit[18] of the confluent Heun equation as a particular case achieved by the simple choice ε = 0. We discuss here three different sets of such relations applying to the Kummer confluent hypergeometric functions of the form 1 F1(α0 + n; γ0 + n; s0z), 1 F1(α0 + n; γ0; s0z) and 1 F1(α0; γ0 + n; s0z), respectively.

EXPANSIONS

DISCUSSION