A Generalized Framework for the (a, b)-Transformation of Probability Measures

In this paper, we propose an analytic deformation acting on probability measures, designed to encompass and extend two fundamental operators in free probability: the (a,b)- and the Tc-deformations. This unified operator, indicated by X(a,b,c), is introduced through a functional relation for the Cauchy–Stieltjes transform. We have X(a,b,0)=U˜(a,b) and X(1,1,c)=Tc. We examine the structural properties of this transformation within the setting of Cauchy–Stieltjes kernel (CSK) families, with special emphasis on the behavior of the associated variance functions (VFs). An explicit formula for the VF corresponding to measure deformed by X(a,b,c) is established. This result allows us to demonstrate a key invariance property: the free Meixner class of probability measures remains stable under the X(a,b,c)-transformation. Furthermore, a novel characterization of the semicircle law is obtained through the action of X(a,1,c), highlighting the role of symmetry in the deformation and preservation of free-probabilistic distributions.