Asymptotic behaviour of the Urbanik semigroup

We revisit the product convolution semigroup of probability densities ec(t),c>0 on the positive half-line with moments (n!)c and determine the asymptotic behaviour of ec for large and small t>0. This shows that (n!)c is indeterminate as Stieltjes moment sequence if and only if c>2. When c is a natural number ec is a Meijer-G function. From the results about ec we obtain the asymptotic behaviour at ±∞ of the convolution roots of the Gumbel distribution.