A two-parameter extension of the Urbanik semigroup

Abstract

We prove that s_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\le 2. The latter was conjectured in a paper by Lin (ArXiv: 1711.01536) 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 behaviour 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 previous work of the author and J. L. L\'opez 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.