Elementary integral series for Heun functions: Application to black-hole perturbation theory

Heun differential equations are the most general second order Fuchsian with four regular singularities. An explicit integral series representation of functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations solutions all class: general, confluent, bi-confluent, doubly triconfluent. All illustrated concrete examples use, Python implementations available for download. demonstrate utility by providing first solution to Teukolsky radial equation governing metric perturbations rotating black holes that is convergent everywhere from hole horizon up spatial infinity.