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In a recent paper, the canonical forms of a new multi-parameter class of Abeldifferential equations, so-called AIR, all of whose members can be mapped intoRiccati equations, were shown to be related to the differential equations forthe hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connectionbetween the AIR canonical forms and the Heun General (GHE), Confluent (CHE) andBiconfluent (BHE) equations is presented. This connection fixes the value ofone of the Heun parameters, expresses another one in terms of those remaining,and provides closed form solutions in terms of pFq functions for the resultingGHE, CHE and BHE, respectively depending on four, three and two irreducibleparameters. This connection also turns evident what is the relation between theHeun parameters such that the solutions admit Liouvillian form, and suggests amechanism for relating linear equations with N and N-1 singularities throughthe canonical forms of a non-linear equation of one order less.

Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations