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

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.

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

“…Substituting n R and 2 n P  from equations (31), (33) and cancelling the common denominator, we obtain the following polynomial equation in n : Since the terms proportional to 2 N n  cancel, this equation is of degree 1 N  . The point now is that the coefficient of the remaining highest-order term proportional to 1 N n  is checked to be equal to  .…”

Heun functions related to entropies

“…Let us consider the Heun function Hl 1 2 , q; 2q, 1; 1, 1; x , which was studied also in the preceding section. The parameters α = 2q, β = 1, γ = 1, δ = 1, ǫ = 2q satisfy the condition (33) from [9], namely a = 1 2 , γ + δ = 2, q = aαβ + a(1 − δ)ǫ.…”

“…Our main results are concerned with the closed forms of the functions Hl 1 2 , −2nθ; −2n, 2θ; γ, γ; x and Hl 1 2 , 2nθ; 2n, 2θ; γ, γ; x which generalize the functions F n (x), respectively G n (−x). Section 3 is devoted to the representation of Hl 1 2 , q; 2q, 1; 1, 1; x in terms of hypergeometric functions, in the spirit of [9].…”

Two-state model of a general Heun class with periodic level-crossings

“…However, during the past years a progress is recorded following the approach suggested by Svartholm [22] and Erdélyi [23]. Several new series expansions of the general Heun function have been constructed in terms of simpler special functions such as the incomplete Beta function, the Gauss hypergeometric function, the Appell generalized hypergeometric function of two variables [24][25][26]. Below we use a specific expansion of the general Heun function which is applicable if a characteristic exponent of the singularity at infinity is zero [24].…”