  We present an Almansi-type decomposition for polynomials with Clifford
coefficients, and more generally for slice-regular functions on Clifford
algebras. The classical result by Emilio Almansi, published in 1899, dealt with
polyharmonic functions, the elements of the kernel of an iterated Laplacian.
Here we consider polynomials of the form $P(x)=\sum_{k=0}^d x^ka_k$, with
Clifford coefficients $a_k\in\mathbb R_{n}$, and get an analogous decomposition
related to zonal polyharmonics. We show the relation between such decomposition
and the Dirac (or Cauchy-Riemann) operator and extend the results to
slice-regular functions.
