Abstract: For the class of free-infinitely divisible transforms we introduce three families of increasing Urbanik type subclasses. 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 random integral representations are known. Special functions like Hurwitz--Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.

2010 AMS Mathematics Subject Classification: Primary 60E07, 60E10, 60H05 ; Secondary 33B15, 33C05.

Keywords and phrases: characteristic function, infinite divisibility, selfdecomposability, s-selfdecomposability, Lévy--Khinchin formula, free-infinite divisibility, Nevanlinna--Pick functions, polygamma function, hypergeometric function.