Elliptic Differential-Difference Equations of General Form in the Half-Space

We study the Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential operators and shift operators not bound by commensurability conditions for shifts. For this problem, we establish classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data), construct an integral representation of the solution by means of a Poisson-type formula, and prove that it approaches to zero as the time-like independent variable tends to infinity.