On Generalized t-Transformation of Free Convolution

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a∈R and b>0. In this article, some results of t-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.