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 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.