Linear ODEs, singularities and non-local transformations
In a recent paper and talk at CECM, a relation between non-linear first order Abel type ODEs and linear ODEs with four regular singular points (Heun type) and confluent cases was shown. That connection leads to solutions expressed as exponentials of integrals with hypergeometric integrand to three multi-parameter Heun type families. In this talk, an alternative approach, exploring non-local transformations and some tricky properties of linear ODEs, is presented. This approach shortcuts the Abel equation intermediate step entirely and leads to solutions that are free of integrals, for the same three families of Heun equations. Each independent solution is a linear combination of hypergeometric functions such that the linear combination itself is not hypergeometric nor Liouvillian. These solutions are new and are the most general closed form solutions known at present for Heun equations.