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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.

An inequality involving the gamma and digamma functions

Qi F., Guo B.-N.

Journal of Applied Analysis

2016

Vol. 22, nr 1

49--54

In the paper, the authors establish an inequality involving the gamma and digamma functions and apply it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.