Heun polynomials and exact solutions for the massless Dirac particle in the C-metric

The equation of motion of a massless Dirac particle in the C-metric leads to the general Heun equation (GHE) for the radial and the polar variables. The GHE, under certain parametric conditions, has been cast in terms of a new set of $su(1,1)$ generators involving differential operators of \emph{degrees} $\pm 1/2$ and $0$. Additional \emph{Heun polynomials} are obtained using this new algebraic structure and are used to construct some exact solutions for the radial and the polar parts of the Dirac equation.