  We consider the set of Stieltjes moment sequences, for which every positive
power is again a Stieltjes moment sequence, we and prove an integral
representation of the logarithm of the moment sequence in analogy to the
L\'evy-Khinchin representation. We use the result to construct product
convolution semigroups with moments of all orders and to calculate their Mellin
transforms. As an application we construct a positive generating function for
the orthonormal Hermite polynomials.
