  It is very difficult and important to construct Bell inequalities for
n-partite, k-settings of measurement, and d-dimensional (n,k,d) systems.
Inspired by the iteration formula form of the
Mermin-Ardehali-Belinski{\u{\i}}-Klyshko (MABK) inequality, we generalize the
multi-component correlation functions for bipartite d-dimensional systems to
n-partite ones, and construct the corresponding Bell inequality. The
Collins-Gisin-Linden-Massar-Popescu inequality can be reproduced by this way.
The most important result is that for prime d the general Bell function in
full-correlated multi-component correlation function form for (n,2,d) systems
can be reformulated in iteration formula by two full-correlated multi-component
Bell functions for (n-1,2,d) systems. As applications, we recover the MABK
inequality and the most robust coincidence Bell inequalities for
(3,2,3),(4,2,3),(5,2,3), and (3,2,5) Bell scenarios with this iteration
formula. This implies that the iteration formula is an efficient way of
constructing multi-partite Bell inequalities. In addition, we also give some
new Bell inequalities with the same robustness but inequivalent to the known
ones.
