Free probability theory was introduced by D. Voiculescu in the 1980s. The notion of free independence, an analogue of classical independence, provides a powerful tool for studying random matrix theory. The S-transform is one of the main tools used to analyze the distribution of the product of freely independent random variables, which is known as free multiplicative convolution.
In 2007, K. J. Dykema introduced the concept of non-crossing linked partitions, which provides a recurrence formula for computing the coefficients of the inverse of the S-transform.
In this talk, we will first briefly review this background. Then, we will introduce the concept of infinitesimal freeness, a generalization of free probability, and discuss how Dykema’s results can be extended to the infinitesimal setting.