Linear ODEs, singularities and non-local transformations

Edgardo Cheb-Terrab

July 19th, 2004.

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.