The objective of this work is to investigate the T=(s,t)-deformed free convolution ⊞T for s>0 and t∈R and to clarify its structural and asymptotic properties within the framework of Cauchy–Stieltjes kernel (CSK) families. The methodology is based on the analysis of the associated variance functions (VFs), which provide an effective analytic tool for describing deformation mechanisms, invariance properties, and convolution structures. In particular, we derive an explicit formula for the VF of convolution powers and exploit this representation to develop approximation procedures for distributions in CSK families generated by the (s,t)-deformed free Gaussian and free Poisson laws. We also establish several limit theorems describing the asymptotic behavior of the deformation. These findings highlight intrinsic symmetry and scaling properties and reveal connections with free additive, Boolean additive, and free multiplicative convolutions, thereby placing the (s,t)-deformation within a unified probabilistic framework governed by transformation, invariance, and structural regularity.
Fakhfakh et al. (Mon,) studied this question.