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2 Jurek Zbigniew J.

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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.

On a Relation Between Classical and Free Infinitely Divisible Transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

349--367

We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.