Affine relation between an infinitely divisible distribution function and its Lévy measure

A certain subfamily of discrete compound Poisson laws is known to be characterised in terms of an affine relation linking the probability masses with those of its jump law. We show that the continuous version of this relation leads to the so-called Geeta laws and the Kendall-Ressel law (for the zero hitting time of a drifted gamma process). This yields an explicit expression for the Laplace transform of the Kendall-Ressel law in terms of the secondary branch of the Lambert function and it offers an alternative perspective on a recent treatment of the gamma process hitting time. The ambit of logarithmic complete monotonicity of a certain family of functions is extended.