Catalan Numbers and Free Distributions of Mutually Free Multi Semicircular Elements

In this paper, we consider certain relations among Catalan numbers, and study free distributions of mutually free, multi semicircular elements. We not only characterize the joint free distributions of given multi semicircular elements, but also provide estimations for such free-distributional data by using our relations among Catalan numbers. As application, asymptotic behaviors of the joint free distribution of mutually free, multi semicircular elements are considered.