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We study two ways (levels) of finding free-probability analogues of classicalinfinitely divisible measures. More precisely, we identify their Voiculescutransforms. For free-selfdecomposable measures we found the formula (adifferential equation) for their background driving transforms. We illustrateour methods on the hyperbolic characteristic functions. As a by-product ourapproach may produce new formulas for some definite integrals.

Ona relation between classical and free infinitely divisible transforms