Classes of exact solutions for the massless Dirac particle in the $C$-metric

The massless Dirac particle in the $C$-metric, representing the exterior gravitational field of a uniformly accelerating black hole, is studied. Classes of (quasi-)polynomial solutions to the radial and the polar parts of the Dirac equation, each of which is equivalent to the general Heun equation~(GHE), are obtained exploiting the underlying $su(1,1)$ algebraic structures of the GHE.