Generalized confluent hypergeometric solutions of the Heun confluent equation

Abstract

We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.

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

“…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]. The results are in agreement with the conjecture by Takemura [18] which suggests that generalized-hypergeometric solutions exist for any Fuchsian differential equation having 3 M + regular singularities of which M are apparent [25].…”

“…We note that the series (6) with such coefficients may terminate only if α or β is a nonpositive integer (besides, 0 θ and 1 θ should obey certain polynomial equations). Having this in the mind and based on the experience gained in treating the general and single-confluent Heun equations [11,12], we try the ansatz (5) for 2 r N = + , 1 s N = + , and ,...., , , ,...,…”

“…The singularities are located at 1 0,1, , z a a = and at z = ∞ . The regularity of the singularity at infinity is ensured by the Fuchsian condition 1 1 α β γ δ ε ε (1) in terms of the generalized hypergeometric functions [9,10], we apply the approach developed in [11,12] for Heun-type equations which originate, via coalescence of singularities, from the general Heun equation [6][7][8]. 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.…”

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

“…The expansion applies if  is not zero,    is not zero or a negative integer and   Our result is that this recurrence admits two-term reductions for infinitely many particular choices of the involved parameters. These reductions are achieved by the following ansatz guessed from the results of [18]:…”

“…confluent Heun equation presented in[18]. There exist expansions of the solutions of the confluent Heun equation in terms of the Kummer confluent hypergeometric functions the form of which differs from those applied in expansion(2).…”