nLab Heun equation

Heun equation is a second order linear ordinary differential equation of Fuchsian type with 4 regular singular points at $0,1,a,\infty$. Its normalized holomorphic solution is the Heun function. If some of the singular points coalesce we talk about the confluent Heun equation.

• wikipedia Heun function

An application of monodromy analysis of Heun equation (coming from separation of variables) to the physics of black holes:

• F. Novaes, Bruno Carneiro da Cunha, Isomonodromy, Painlevé transcendents and scattering off of black holes, J. High Energ. Phys. 2014, 132 (2014) doi arXiv:1404.5188
• Bruno Carneiro da Cunha, João Paulo Cavalcante, Confluent conformal blocks and the Teukolsky master equation, Phys. Rev. D 102, 105013 (2020) doi arXiv:1906.10638; Teukolsky master equation and Painlevé transcendents: numerics and extremal limit, arXiv:2105.08790