  We introduce and study the notion of k-divisible elements in a
non-commutative probability space. A k-divisible element is a (non-commutative)
random variable whose n-th moment vanishes whenever n is not a multiple of k.
First, we consider the combinatorial convolution \ast in the lattices NC of
non-crossing partitions and NC^k of k-divisible non-crossing partitions and
show that convolving k times with the zeta-function in NC is equivalent to
convolving once with the zeta-function in NC^k. Furthermore, when x is
k-divisible, we derive a formula for the free cumulants of x^k in terms of the
free cumulants of x, involving k-divisible non-crossing partitions. Second, we
prove that if a and s are free and s is k-divisible then sps and a are free,
whenever p is any polynomial (on a and s) of degree k - 2 on s. Moreover, we
define a notion of R-diagonal k-tuples and prove similar results. Next, we show
that free multiplicative convolution between a measure concentrated in the
positive real line and a probability measure with k-symmetry is well defined.
Analytic tools to calculate this convolution are developed. Finally, we
concentrate on free additive powers of k-symmetric distributions and prove that
\mu t is a well defined probability measure, for all t > 1. We derive central
limit theorems and Poisson type ones. More generally, we consider freely
infinitely divisible measures and prove that free infinite divisibility is
maintained under the mapping \mu \rightarrow \mu^k . We conclude by focusing on
(k-symmetric) free stable distributions, for which we prove a reproducing
property generalizing the ones known for one sided and real symmetric free
stable laws.
