In this paper, we introduce two analytic deformations of probability measures that unify and extend two classical deformations from free probability theory, namely the T=(s,t)-deformation UT and the Ta-deformation, where a,t∈R and s>0. The corresponding operators, denoted by Y(a,s,t)′ and Y(a,s,t)′′, are defined via a functional equation involving the Cauchy–Stieltjes transform (CST). This framework recovers the classical cases as particular instances, specifically Y(0,s,t)′=Y(0,s,t)′′=UT and Y(a,1,1)′=Y(a,1,1)′′=Ta. We analyze the analytic and structural properties of the operators Y(a,s,t)′ and Y(a,s,t)′′ within the concept of Cauchy–Stieltjes kernel (CSK) families, with particular emphasis on their action on variance functions (VFs). In particular, we derive explicit formulas for the VFs associated with measures deformed by Y(a,s,t)′ and Y(a,s,t)′′. As an application, we establish an invariance property showing that the class of free Meixner family (FMF) is stable under both deformations. Furthermore, by restricting the parameters to Y(a,1,t)′ and Y(a,1,t)′′, we obtain two new characterizations of the semicircle law. These results highlight the role of symmetry in the analytic deformation and in the stability properties of fundamental distributions in free probability.
Fakhfakh et al. (Fri,) studied this question.