  We propose a natural family of higher-order partial differential equations
generalizing the second-order Klein-Gordon equation. We characterize the
associated model by means of a generalized action for a scalar field,
containing higher-derivative terms. The limit obtained by considering
arbitrarily higher-order powers of the d'Alembertian operator leading to a
formal infinite-order partial differential equation is discussed. The general
model is constructed using the exponential of the d'Alembertian differential
operator. The canonical energy-momentum tensor densities and field propagators
are explicitly computed. We consider both homogeneous and non-homogeneous
situations. The classical solutions are obtained for all cases.
