The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman
  black hole spacetimes
  Exact solutions of the Dirac general relativistic equation (DE) that describe
the dynamics of a massive, electrically charged particle with half-integer spin
in the curved spacetime geometry of an electrically charged, rotating
Kerr-Newman-(anti) de Sitter black hole (BH) are investigated. We first, derive
the DE in the Kerr-Newman-de Sitter (KNdS) BH background using a generalised
Kinnersley null tetrad in the Newman-Penrose formalism. In this frame, we prove
the separation of the DE into ordinary differential equations for the radial
and angular parts. Under specific transformations of the independent and
dependent variables we prove that the transformed radial equation for a massive
charged spin $\frac{1}{2}$ fermion in the background KNdS BH constitutes a
highly non-trivial generalisation of Heun's equation. Using a
Regge-Wheeler-like independent variable we transform the radial equation in the
KNdS background into a Schr\"{o}dinger-like differential equation and
investigate its asymptotic behaviour near the event and cosmological horizons.
For a massive fermion (MF) in the background of a Kerr-Newman (KN) BH we first
prove that the radial and angular equations that result from the separation of
DE reduce to the generalised Heun differential equation (GHE). The local
solutions of such GHE are derived and can be described by holomorphic functions
whose power series coefficients are determined by a four-term recurrence
relation. Using asymptotic analysis we derive the solutions for the MF far away
from the KN BH and the solutions near the event horizon . The determination of
the separation constant as an eigenvalue problem in the KN background is
investigated. Using the aforementioned four-term recursion formula we prove
that in the non-extreme KN geometry there are no bound states with
$\omega^2<\mu^2$, where $\omega$ and $\mu$ are the energy and mass of the
fermion respectively.
