New solutions for the General, Confluent and Bi-Confluent Heun equations and their connection with Abel equations

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection exists after fixing the value of one of the Heun parameters and expressing another one in terms of those remaining. The resulting GHE, CHE and BHE families respectively depend on four, three and two irreducible parameters. This connection provides closed form solutions in terms of pFq functions for these Heun equation families, shows that the problems formulated in terms of Abel AIR equations can also be formulated in terms of these linear GHE, CHE and BHE equations, and suggests a mechanism for relating linear equations with N and N-1 singularities. Introduction The Heun equation [1] is a second order linear equation of the form y′′ + ( γ x + δ x− 1 + ǫ x− a ) y′ + αβ x− q x (x− 1) (x− a) y = 0, (1) where {α, β, γ, δ, ǫ, a, q} are constant with respect to x, are related by γ+ δ+ ǫ = α+β+1, and a 6= 0, a 6= 1. This equation has four regular singular points, at {0, 1, a,∞}. Through confluence processes, equation (1), herein called the General Heun Equation (GHE), transforms into four other multi-parameter equations [2], so-called Confluent (CHE), Biconfluent (BHE), Doubleconfluent (DHE) and Triconfluent (THE). Through transformations of the form y → P (x) y, these five equations can be written in normal form, using the notation of [2], in terms of arbitrary constants {a, A,B,C,D,E, F}; for the 6-parameter GHE (1) we have y′′ + ( A x + B x− 1 − A+B x− a + D x2 + E (x− 1) + F (x− a) ) y = 0 (2) The 5-parameter CHE,