Solutions for the General, Confluent and Biconfluent 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 fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less. 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) )