A Common Generalization of the (a,b)- and (s,t)-Transformations of Probability Measures

This paper presents two analytic mappings defined on probability measures that extend and unify the (a,b)- and (s,t)-deformations arising in free probability for s, b>0 and a, t∈R. These unified operators, denoted U(a,b,s,t)′ and U(a,b,s,t)′′, are characterized by a functional equation involving the Cauchy–Stieltjes transform, providing a transform-based formulation of measure deformation. They reduce to the (a,b)-transformation when s=t=1 and to the (s,t)-transformation when a=b=1. Working in the framework of Cauchy–Stieltjes kernel families, we study the induced effect of these transformations on the associated variance functions and obtain explicit transformation formulas. These results yield a stability theorem showing that the free Meixner class is stable under both operators. In addition, we derive two properties of the semicircle law via the restricted deformations U(a,b,1/b,t)′ and U(a,b,1/b,t)′′, thereby emphasizing the structural role of symmetry in measure transformations and in the preservation of canonical measures.