We generalize the infinitesimal independence appearing in free probability of type B in two directions: to higher order derivatives and other natural independences: tensor, monotone and Boolean. Such generalized infinitesimal independences can be defined by using associative products of infinitely many linear functionals, and therefore the associated cumulants can be defined. These products can be seen as the usual natural products of linear maps with values in formal power series.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-6,
author = {Takahiro Hasebe},
title = {Differential independence via an associative product of infinitely many linear functionals},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {79-94},
zbl = {1235.46062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-6}
}
Takahiro Hasebe. Differential independence via an associative product of infinitely many linear functionals. Colloquium Mathematicae, Tome 122 (2011) pp. 79-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-6/