On a relation between classical and free infinitely divisible transforms

We study two ways (levels) of finding free-probability analogue of classical infinite divisible measures. More precisely, we identify their Voiculescu transforms. For free - selfdecomposable Voiculescu transform we found the formula (a differential equation) for their background driving transforms. We illustrate our methods on the selfdecomposable hyperbolic characteristic functions. As a by-product our approach potentially may produce new formulas for definite integrals of some special functions. (cf. www.arXiv:1707.02540v1 [math.PR])