  We define higher order infinitesimal noncommutative probability space and
infinitesimal non-crossing cumulant functionals. In this framework, we
generalize to higher order the notion of infinitesimal freeness, via a
vanishing of mixed cumulants condition. We also introduce and study some
non-crossing partitions related to this notion. Finally, as an application, we
show how to compute the successive derivatives of the free convolution of two
time-indexed families of distributions from their individual derivatives.
