Natural higher-derivatives generalization for the Klein-Gordon equation

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.