  We investigate polyharmonic functions associated to Brownian motion and
random walks in cones. These are functions which cancel some power of the usual
Laplacian in the continuous setting and of the discrete Laplacian in the
discrete setting. We show that polyharmonic functions naturally appear while
considering asymptotic expansions of the heat kernel in the Brownian case and
in lattice walk enumeration problems. We provide a method to construct general
polyharmonic functions through Laplace transforms and generating functions in
the continuous and discrete cases, respectively. This is done by using a
functional equation approach.
