On Generalized Va-Transformation of Measures

In this study, we introduce a novel transformation of probability measures that unifies two significant transformations in free probability theory: the t-transformation and the Va-transformation. Our unified transformation, denoted U(a,t), is defined analytically via a modified functional equation involving the Cauchy transform, and reduces to the t-transformation when a=0, and to the Va-transformation when t=1. We investigate some properties of this new transformation from the lens of Cauchy–Stieltjes kernel (CSK) families and the corresponding variance functions (VFs). We derive a general expression for the VF resulting from the U(a,t)-transformation. This new expression is applied to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, novel limiting theorems involving U(a,t)-transformation are proved providing new insights into the relations between some important measures in free probability such as the semicircle, Marchenko–Pastur, and free binomial measures.