On a New Extension of the t-Transformation of Probability Measures

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by T(a,t), which unifies the classical t- and Ta-transformations in free probability. We derive the functional equation characterizing T(a,t) through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when a=0 or t=1. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under T(a,t). Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws.