  We present explicit formulas for solutions to nonhomogeneous boundary value
problems involving any positive power of the Laplacian in the half-space. For
non-integer powers the operator becomes nonlocal and this requires a suitable
extension of Dirichlet-type boundary conditions. A key ingredient in our proofs
is a point inversion transformation which preserves harmonicity and allows us
to use known results for the ball. We include uniqueness statements, regularity
estimates, and describe the growth or decay of solutions at infinity and at the
boundary.
