  In this paper, we are concerned with equations \eqref{PDE} involving
higher-order fractional Laplacians. By introducing a new approach, we prove the
super poly-harmonic properties for nonnegative solutions to \eqref{PDE}
(Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem.
As a consequence, we derive many important applications of the super
poly-harmonic properties. For instance, we establish Liouville theorems,
integral representation formula and classification results for nonnegative
solutions to fractional higher-order equations \eqref{PDE} with general
nonlinearities $f(x,u,Du,\cdots)$ including conformally invariant and odd order
cases. In particular, our results completely improve the classification results
for third order equations in Dai and Qin \cite{DQ1} by removing the assumptions
on integrability. We also derive a characterization for $\alpha$-harmonic
functions via averages in the appendix.
