Perturbative and semi-analytical solutions to Teukolsky equations for massive fermions

In this work, we aim at solving the Teukolsky equations for a fermion with mass $m_e\mathrm{\ne }0$ in the presence of a rotating black hole with mass $M$. We consider two different regimes: ${\tilde{m}}_e=M^{-1}{m}_e\ll 1$ and $a\omega \ll 1$; ${\tilde{m}}_e\ll 1$ and $a\omega \gtrsim 1$. We treat each of these two regimes in different ways: we use a perturbative approach for the first, similar to the \emph{usual} one employed for spin 0, 1, 2 and mass-less 1/2 fields, but with two small parameters (a$\omega$ and ${\tilde{m}}_e$); as we shall see, the second can be treated with a semi-analytical approach. In a forthcoming paper we shall study the remaining two cases in which ${\tilde{m}}_e\gtrsim 1$, while $a\omega \ll 1$ or $a\omega \gtrsim 1$. The regime with ${\tilde{m}}_e\ll 1$, but $a\omega \gtrsim 1$ is probably the most interesting from the astrophysics point of view, but this last two cases might be of some interest for the study of the interaction of fermions with very small black holes, which may be formed, for example, in the last stages of the Hawking evaporation.