  We compute new exact analytic expressions for one-loop scalar effective
actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions.
These are computed by the connection coefficients of the Heun equation via a
generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with
regular singularities. The expressions we find are in terms of
Nekrasov-Shatashvili special functions, making explicit the analytic properties
of the one-loop effective actions with respect to the gravitational parameters
and the precise contributions of the quasi-normal modes. The latter arise via
an associated integrable system. In particular, we prove asymptotic formulae
for large angular momenta in terms of hypergeometric functions and give a
precise mathematical meaning to Rindler-like region contributions. Moreover we
identify the leading terms in the large distance expansion as the point
particle approximation of the BH and their finite size corrections as encoding
the BH tidal response. We also discuss exact properties of the thermal version
of the BH effective actions. Although we focus on the real scalar field in
dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae
can be given for higher spin matter and radiation fields in more general
gravitational backgrounds.
