Notes on Cauchy–Stieltjes Kernel Families - Shokrya S. Alshqaq; Raouf Fakhfakh; Fatimah Alshahrani (Axioms)
The free Meixner family (<semantics>FMF</semantics>) 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 <semantics>≤2</semantics> 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 <semantics>Ta</semantics>-transformation of probability measures. Denote by <semantics>P</semantics> the set of (non-degenerate) real probabilities. For <semantics>ν∈P</semantics> and <semantics>a∈R</semantics>, consider the transformation of measure <semantics>ν</semantics>, denoted <semantics>Ta(ν)</semantics>, defined by <semantics>FTa(ν)(w)=Fν(w−a)+a,</semantics> where <semantics>Fν(·)</semantics> is the inverse of the Cauchy–Stieltjes transformation of <semantics>ν</semantics>. In this study, we provide important insights into the notion of the <semantics>Ta</semantics>-transformation of probabilities. We demonstrate that the <semantics>FMF</semantics> (respectively, the free counterpart of the Letac–Mora class of measures) is invariant under the <semantics>Ta</semantics>-transformation. Furthermore, we develop additional characteristics of the <semantics>Ta</semantics>-transformation, which yield intriguing findings for significant free probability distributions such as the free Poisson and free Gamma distributions.