Summary: We prove thats \(n(a, b) = \Gamma (an + b)/\Gamma(b), n = 0, 1,\ldots\), is an infinitely divisible Stieltjes moment sequence for arbitrary \(a, b > 0\). Its powers \(s_n(a,b)^c, c > 0\), are Stieltjes determinate if and only if \(ac \leqslant 2\). The latter was conjectured in a paper by Lin (2019) in the case \(b =1\). We describe a product convolution semigroup \(\tau_c(a, b), c > 0\) of probability measures on the positive half-line with densities \(e_c(a, b)\) and having the moments \(s_n(a,b)^c\). We determine the asymptotic behavior of \(e_c(a, b)(t)\) for \(t \to 0\) and for \(t \to \infty\), and the latter implies the Stieltjes indeterminacy when \(ac > 2\). The results extend the previous work of the author and and J. L. López [J. Approx. Theory 195, 109–121 (2015; Zbl 1317.30058)] and lead to a convolution semigroup of probability densities \((g_c(a,b)(x))_{c>0}\) on the real line. The special case \((g_c(a,1)(x))_{c<0}\) are the convolution roots of the Gumbel distribution with scale parameter \(a > 0\). All the densities \(g_c(a,b)(x)\) lead to determinate Hamburger moment problems.