A Study of the (a, b)-Deformed Free Convolution

This study is devoted to the detailed examination of the concept of (a,b)-deformation, defined for parameters a∈R and b>0. The analysis is conducted within the framework of Cauchy–Stieltjes kernel (CSK) families of probability measures, with particular attention given to the role of their variance functions (VFs). Using the VF as the main analytical tool, it is shown that the (a,b)-deformation of any measure belonging to the free Meixner family (FMF) remains within the same family. Moreover, the VF framework provides a powerful and flexible means for establishing new limit theorems associated with (a,b)-deformed free convolution. In particular, several novel limiting behaviors are derived, which naturally encompass both free and Boolean additive convolutions as special cases.