Notes on Cauchy–Stieltjes Kernel Families

The free Meixner family (FMF) is the family of measures that produces quadratic Cauchy–Stieltjes Kernel (CSK) families (i.e., meaning that the associated variance function (VF) is a polynomial with degree ≤2 in the mean). Furthermore, a cubic class is introduced in the context of CSK families and is connected to the quadratic class via a reciprocity relation. The associated probability measures are the so-called free analog of the Letac–Mora class (with VF of degree 3). In free probability theory, these two classes of probabilities are crucial. However, a novel transformation of measures is introduced in the setting of free probability, known as the Ta-transformation of probability measures. Denote by P the set of (non-degenerate) real probabilities. For ν∈P and a∈R, consider the transformation of measure ν, denoted Ta(ν), defined by FTa(ν)(w)=Fν(w−a)+a, where Fν(·) is the inverse of the Cauchy–Stieltjes transformation of ν. In this study, we provide important insights into the notion of the Ta-transformation of probabilities. We demonstrate that the FMF (respectively, the free counterpart of the Letac–Mora class of measures) is invariant under the Ta-transformation. Furthermore, we develop additional characteristics of the Ta-transformation, which yield intriguing findings for significant free probability distributions such as the free Poisson and free Gamma distributions.