Redshift of a photon emitted along the black hole horizon

In this work we derive some general features of the redshift measured by radially moving observers in the black hole background. Let observer 1 cross the black hole horizon emitting of a photon while observer 2 crossing the same horizon later receives it. We show that if (i) the horizon is the outer one (event horizon) and (ii) it is nonextremal, received frequency is redshifted. This generalizes previous recent results in literature. For the inner horizon (like in the Reissner-Nordstr{o}m metric) the frequency is blueshifted. If the horizon is extremal, the frequency does not change. We derive explicit formulas describing the frequency shift in generalized Kruskal- and Lemaitre-like coordinates.