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, the so-called Abel inverse Riccati (AIR), all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric F-2(1), F-1(1), and F-0(1), functions. In this paper, a connection between the AIR canonical forms and the general Heun (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 F-p(q) functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also makes evident 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 nonlinear equation of one order less.