Time-dependent gamma-ray production in the anisotropic inverse Compton e+/- pair cascade initiated by electrons in active galaxies

New high-energy emission features from active galaxies have recently been discovered using the Cherenkov telescopes; for example, a time-scale variability of a few minutes for TeV emission from Mrk 501 and PKS 2155-304, sub-TeV γ-ray emission from the GeV peaked blazar 3C 279, and TeV emission from two nearby active galaxies, M87 and Cen A, whose jets are inclined at a relatively large angle to the line of sight. These results have shed new light on the high-energy processes occurring in the central parts of active galaxies and have stimulated more detailed studies of γ-ray emission models. In this paper, we report on the results of a detailed analysis concerning the most general version of the model for γ-ray production by leptons injected in the jet, which interact with the thermal radiation from an accretion disc: the so-called external inverse Compton (IC) model. We investigate the γ-ray spectra produced in an anisotropic IC e^+/- pair cascade in the whole volume above the accretion disc. The cascade γ-ray spectra are obtained for different locations of the observer with respect to the direction of the jet. We also study the time evolution of this γ-ray emission caused by the propagation of the relativistic leptons along the jet and the delays resulting from different places of the origin of γ-rays above the accretion disc. We discuss the main features of such a cascade model, assuming a constant injection rate of electrons along the jet. We are investigating two models for maximum energies of injected electrons: with a constant value independent of the distance along the jet or limited by the synchrotron energy losses considered locally in the jet. The model is discussed in the context of blazars observed at small and large inclination angles, taking as an example the parameters of the two famous sources, Cen A and 3C 279.