A Two-Parameter Extension of the Va Deformation of Probability Measures

This article proposes a single-deformation scheme for probability measures that simultaneously encompasses two classical deformations from free probability: the Va deformation (a∈R) and the Tc deformation (c∈R). The associated operator, written as W(a,c), is introduced via a functional relation involving the Cauchy–Stieltjes transform and is constructed so as to recover the initial deformations as special cases, namely W(0,c)=Tc and W(a,0)=Va. Working within the concept of Cauchy–Stieltjes kernel families, we analyze the action of W(a,c) on variance functions and establish an explicit expression for the variance function induced by this deformation. This approach leads to a structural invariance property demonstrating that the free Meixner class is preserved under the action of W(a,c). In addition, the operator provides a new perspective on the semicircle distribution, yielding a characterization that reflects the symmetric nature of the deformation and its compatibility with fundamental distributions in free probability.