A two-parameter extension of Urbanik’s product convolution semigroup

We prove that sna, b = Γan + b/Γb, n = 0, 1, . . ., is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers sna, bc, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin 2019 in the case b = 1. We describe a product convolution semigroup τca, b, c > 0, of probability measures on the positive half-line with densities eca, b and having the moments sna, bc. We determine the asymptotic behavior of eca, bt for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and Lopez 2015 and lead to a convolution semigroup of probability densities gca, bxc>0 on the real line. The special case gca, 1xc>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gca, bx lead to determinate Hamburger moment problems