Stability and properties of Cauchy–Stieltjes Kernel families under generalized $t$-transformation

The study of the generalized $t$-deformation of free convolution from the lens of Cauchy–Stieltjes kernel (CSK) families provides an excellent mathematical framework for understanding noncommutative probability distributions. In this paper, we use the concept of generalized $t$-deformation to demonstrate different elements of the Marchenko–Pastur, free Gamma, and inverse semicircle measures in the CSK families setting. These findings advance our knowledge of generalized $t$-deformation in the non-commutative probability framework.