Free Probability for Pairs of Faces III: 2-Variables Bi-free Partial S- and T-Transforms

TL;DR

This paper introduces two new 2-variable transforms, the partial bi-free S-transform and T-transform, which extend the framework of bi-free convolutions for pairs of variables, building on previous work with the bi-free R-transform.

Contribution

It presents the definitions and properties of the partial bi-free S- and T-transforms, expanding the toolkit for analyzing bi-free convolutions of pairs of variables.

Findings

Defines the partial bi-free S-transform and T-transform

Establishes their properties and relations to bi-free convolutions

Extends previous bi-free transform frameworks

Abstract

We introduce two 2-variables transforms: the partial bi-free S-transform and the partial bi-free T-transform. These transforms are the analogues for the bi-multiplicative and respectively for the additive-multiplicative bi-free convolution of the 2-variables partial bi-free R-transform in our previous paper in this series.