Urbanik type subclasses of the free-infinitely divisible transforms

TL;DR

This paper introduces three new subclasses of free-infinitely divisible transforms, extending from free-normal to the entire class, using integral representations involving special functions.

Contribution

It defines Urbanik type subclasses for free-infinitely divisible transforms based on classical measure analogs and their integral representations.

Findings

Three families of subclasses are introduced.

Special functions appear in the integral kernels.

The subclasses connect free and classical infinitely divisible measures.

Abstract

For the class of free-infinitely divisible transforms are introduced three families of increasing Urbanik type subclasses of those transforms. They begin with the class of free-normal transforms and end up with the whole class of free-infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which are known their random integral representations. Special functions like Hurwitz-Lerch, polygamma and hypergeometric appeared in kernels of the corresponding integral representations.