The formal series Witt transform

Given a formal power series f(z)∈C〚z〛 we define, for any positive integer r, its rth Witt transform, Wf(r), by Wf(r)(z)=1r∑d|rμ(d)f(zd)r/d, where μ denotes the Möbius function. The Witt transform generalizes the necklace polynomials, M(α;n), that occur in the cyclotomic identity11-αy=∏n=1∞(1-yn)-M(α;n).Several properties of Wf(r) are established. Some examples relevant to number theory are considered.