  The construction of a simplicial complex given by polyhedral joins
(introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and
Gitler's $J$-construction and simplicial wedge construction. This article gives
a cohomological decomposition of a polyhedral product over a polyhedral join
for certain families of pairs of simplicial complexes. A formula for the
Hilbert-Poincar\'{e} series is given, which generalizes Ayzenberg's formula for
the moment-angle complex.
