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Urbanik type subclasses of free-infinitely~divisible~transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

23--39

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 repre- sentations are known. Special functions like Hurwitz–Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.