Bt-Transformation and Variance Function

This study investigates the Bt-transformation of probability measures within the framework of free probability. A primary focus is the invariance under this transformation of two fundamental families: the free Meixner family and the free analog of the Letac–Mora class. In addition, we introduce novel characteristics associated with the Bt-transformation, offering refined analytical tools to probe its structural and functional properties. These tools allow us to uncover new and significant properties of several distributions in free probability, including the semicircle, the Marchenko–Pastur, and the free Gamma laws, yielding explicit invariance results and stability conditions. Our findings extend the theoretical understanding of the Bt-transformation and provide practical methods for analyzing the dynamics and stability of classical free distributions under this operator.