A combinatorial approach to Voiculescu's bi-free partial transforms

Paul Skoufranis

Vol. 283 (2016), No. 2, 419–447

DOI: 10.2140/pjm.2016.283.419

Abstract

We present a combinatorial approach to the 2-variable bi-free partial $S$- and $T$-transforms recently discovered by Voiculescu. This approach produces an alternate definition of said transforms using $\left(\ell ,r\right)$-cumulants.

Keywords

bi-free probability, partial bi-free $S$-transform, partial bi-free $T$-transform, bi-free convolutions

Mathematical Subject Classification 2010

Primary: 46L53, 46L54

Milestones

Received: 12 June 2015

Accepted: 8 February 2016

Published: 22 June 2016