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Higher order infinitesimal freeness

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.