A Combinatorial Approach to Voiculescu's Bi-Free Partial Transforms

TL;DR

This paper introduces a combinatorial method to define Voiculescu's bi-free partial S- and T-transforms using $(\,ell, r)$-cumulants, offering an alternative perspective on these mathematical tools.

Contribution

It provides a new combinatorial framework for bi-free partial transforms, expanding the theoretical understanding of these concepts.

Findings

New combinatorial definitions of bi-free partial transforms

Alternative characterization using $(\,ell, r)$-cumulants

Enhanced theoretical insight into bi-free probability

Abstract

In this paper, 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 $(\ell, r)$-cumulants.