Higher order infinitesimal freeness

TL;DR

This paper introduces higher order infinitesimal noncommutative probability spaces and cumulants, generalizing infinitesimal freeness, and provides methods to compute derivatives of free convolutions for time-indexed distributions.

Contribution

It defines higher order infinitesimal freeness, develops related cumulants and partitions, and applies these concepts to compute derivatives of free convolutions.

Findings

Defined higher order infinitesimal noncommutative probability spaces

Generalized infinitesimal freeness to higher orders

Provided a method to compute derivatives of free convolutions

Abstract

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.