A connection between Abel and $_p{\rm F}_q$ hypergeometric differential equations

Abstract

In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions 2F1, 1F1 and 0F1 is unveiled. This connection provides a closed form pFq solution for all Abel equations of the AIR class.