  We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet
problems on balls involving any positive power $s>0$ of the Laplacian. We are
able to prescribe values outside the domain and boundary data of different
orders using explicit Poisson-type kernels and a new notion of higher-order
boundary operator, which recovers normal derivatives if $s$ is a natural
number. Our results unify and generalize previous approaches in the study of
polyharmonic operators and fractional Laplacians. As applications, we show a
novel characterization of $s$-harmonic functions in terms of Martin kernels, a
higher-order fractional Hopf Lemma, and examples of positive and sign-changing
Green functions.
