The formal series Witt transform

Details The formal series Witt transform The formal series Witt transform

2003-11-12

arxiv.org/abs/math/0311194v2

Discrete Mathematics 295 (2005), 143-160

Mathematics

Combinatorics

18 pages, small improvements in contents and presentation, to appear in Discrete Mathematics

Scientific paper

10.1016/j.disc.2005.03.004

Given a formal power series f(z) we define, for any positive integer r, its rth Witt transform, W_f^{(r)}, by rW_f^{(r)}(z)=sum_{d|r}mu(d)f(z^d)^{r/d}, where mu is the Moebius function. The Witt transform generalizes the necklace polynomials M(a,n) that occur in the cyclotomic identity 1-ay=prod (1-y^n)^{M(a,n)}, where the product is over all positive integers. Several properties of the Witt transform are established. Some examples relevant to number theory are considered.

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