  The Green functions of the partial differential operators of even order
acting on smooth sections of a vector bundle over a Riemannian manifold are
investigated via the heat kernel methods. We study the resolvent of a special
class of higher-order operators formed by the products of second-order
operators of Laplace type defined with the help of a unique Riemannian metric
but with different bundle connections and potential terms. The asymptotic
expansion of the Green functions near the diagonal is studied in detail in any
dimension. As a by-product a simple criterion for the validity of the Huygens
principle is obtained. It is shown that all the singularities as well as the
non-analytic regular parts of the Green functions of such high-order operators
are expressed in terms of the usual heat kernel coefficients $a_k$ for a
special Laplace type second-order operator.
