On evaluation of the confluent Heun functions

Abstract

In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points -two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given. arXiv:1804.01007v1 [math.NA] 2 Apr 2018 2 Statement and basic notationsWe use the following form of the confluent Heun equation:This second order linear differential equation has regular singularities at z = 0 and 1, and an irregular singularity of rank 1 at z = ∞ (see e.g. [20]). The parameter q ∈ C is usually referred to as an accessory or auxiliary parameter and γ, δ, ε, α (also belonging to C) are exponent-related parameters.It is important to note that in this paper the parameters ε and α are assumed to be independent. Below, we will use notation c H (q, α, γ, δ, ε; z) or c H (z) for brevity. There are 6 local solutions of equation (1). The Frobenius method can be used to derive local power-series solutions to (1) near z = 0 and z = 1 (two per a singular point), while two solutions at z = ∞ can be obtained in the form of asymptotic series. In § 3 we will present the local solutions near the point z = 0. One of them is analytic in a vicinity of zero and if γ is not a nonpositive integer, we normalize this solution to unity at z = 0 and call it the local confluent Heun function. It is denoted by c Hl (q, α, γ, δ, ε; z). For the second Frobenius local solution, we will use the notation c Hs(q, α, γ, δ, ε; z). When γ is a nonpositive integer, one solution of (1) is analytic in a vicinity of z = 0 but it is equal to zero at z = 0, whereas the second solution can be normalized to unity at zero but generally it is not analytic. Following [16], the normalized solution will be denoted by c Hl (z) and another one by c Hs(z).

Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory

“…which gives the same asymptotic behavior as obtained from series designed to converge when z → +∞ [58,13,62]. The result for p = 0 yields the correct asymptotics of the hypergeometric function obtained in this case.…”

“…which gives the same asymptotic behavior as obtained from series representations of H(z) for z close to 1 [58,13,62].…”

“…We suppose first that p = 0 and assume that δ+γ > 0. In this situation, the confluent Heun function becomes a well understood hypergeometric function [22,58] for which we will nonetheless show that we recover the correct asymptotic behavior. Setting p = 0 we get, as z → +∞, K 1 (z, z 0 ) ∼ a.e 1 + e −(1+4p)z z −γ (z − 1) −δ −e z z γ+δ + e z0 z γ+δ 0 , ∼ a.e e −z z −δ−γ e z0 z γ+δ 0 .…”

Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations

“…The single confluent Heun equation is a remarkable equation that generalizes both the Gauss ordinary and the Kummer confluent hypergeometric equations [1][2][3]. Though this equation has a wide coverage in contemporary physics and mathematics (see, e.g., [1][2][3][4] and references therein; a rather large (yet, not exhaustive) list concerning mainly the general relativity and cosmology is discussed in [5]), and for this reason it has extensively been studied during the recent years (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18]), however, the theory of the equation is currently far from being satisfactory for the most of applications. A reason for this is that the power-series solutions [1][2][3] as well as the expansions of the solutions in terms of simpler special functions (such as the incomplete beta, incomplete gamma, Bessel, Kummer, Coulomb wave, Goursat, Appell functions, etc., see, e.g., [8][9][10][11][12][18][19][20][21][22][23]) are governed by three-or more-term recurrence relations for successive coefficients, so that apart from a few trivial cases the expansion coefficients are not calculated explicitly.…”

Symbolic and numerical analysis in general relativity with open source computer algebra systems

“…Numerical computation of the general and confluent Heun functions are adapted by Oleg V. Motygin for GNU Octave [27,28]. However, no freely available packages or modules can give closed symbolic solutions of these equations.…”