Reduction of singularities in Fuchsian equations, and Heun equations for perturbations of Kerr-Newman-de Sitter and anti-de sitter spacetimes

For Fuchsian differential equations with N singularities at finite points we display two simply expressed conditions on coefficients which guarantee equivalence, under a certain inversion transformation, to a differential equation with N-1 singularities at finite points. For the radial and angular differential equations that govern perturbations of Kerr-Newman-de Sitter and anti-de Sitter spacetimes we find that these two conditions are satisfied identically for arbitrary values of all the parameters in the differential equations, including pole locations, with regular singularities and non-zero cosmological constant. From this we generalize the results of Suzuki, Tarasugi, and Umetsu, Prog. Theor. Phys. 100, 491 (1998) and find 128 Heun equations which are equivalent to the equations which govern the perturbations of the KN(A)dS spacetimes.