Higher order infinitesimal freeness Maxime Fevrier 46L54free probability (of type B)infinitesimal freenessinfinitesimal non-crossing cumulantsnon-crossing partitions 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. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4497 10.1512/iumj.2012.61.4497 en Indiana Univ. Math. J. 61 (2012) 249 - 295 state-of-the-art mathematics http://iumj.org/access/