  We study several kinds of polynomial ensembles of derivative type which we
propose to call P\'olya ensembles. These ensembles are defined on the spaces of
complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and
Hermitian anti-self-dual matrices, and they have nice closure properties under
the multiplicative convolution for the first class and under the additive
convolution for the other classes. The cases of complex square matrices and
Hermitian matrices were already studied in former works. One of our goals is to
unify and generalize the ideas to the other classes of matrices. Here we
consider convolutions within the same class of P\'olya ensembles as well as
convolutions with the more general class of polynomial ensembles. Moreover, we
derive some general identities for group integrals similar to the
Harish-Chandra-Itzykson-Zuber integral, and we relate P\'olya ensembles to
P\'olya frequency functions. For illustration we give a number of explicit
examples for our results.
