  In nature we observe galaxy aggregations that span a wide range of magnitude
gaps between the two first-ranked galaxies of a system ($\Delta m_{12}$). There
are systems with gaps close to zero (e.g., the Coma cluster), and at the other
extreme of the distribution, the largest gaps are found among the so-called
fossil systems. Fossil and non-fossil systems could have different galaxy
populations that should be reflected in their luminosity functions. In this
work we study, for the first time, the dependence of the luminosity function
parameters on $\Delta m_{12}$ using data obtained by the fossil group origins
(FOGO) project. We constructed a hybrid luminosity function for 102 groups and
clusters at $z \le 0.25$. We stacked all the individual luminosity functions,
dividing them into bins of $\Delta m_{12}$, and studied their best-fit
Schechter parameters. We additionally computed a relative luminosity function,
expressed as a function of the central galaxy luminosity, which boosts our
capacity to detect differences, especially at the bright end. We find trends as
a function of $\Delta m_{12}$ at both the bright and faint ends of the
luminosity function. In particular, at the bright end, the larger the magnitude
gap, the fainter the characteristic magnitude $M^\ast$. We also find
differences at the faint end. In this region, the larger the gap, the flatter
the faint-end slope $\alpha$. The differences found at the bright end support a
dissipationless, dynamical friction-driven merging model for the growth of the
central galaxy in group- and cluster-sized halos. The differences in the faint
end cannot be explained by this mechanism. Other processes, such as enhanced
tidal disruption due to early infall and/or prevalence of eccentric orbits, may
play a role. However, a larger sample of systems with $\Delta m_{12} > 1.5$ is
needed to establish the differences at the faint end.
