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