Almansi-type theorems for slice-regular functions on Clifford algebras

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 poly-harmonic 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 $a_kinmathbb R_{n}$, 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.