This paper has four interrelated themes: (1) express Laplace and Mellin transforms of sums of positive random variables in terms of the Mellin transform of the summands; (2) show the equivalence of the two Barnes' lemmas with known properties of gamma distributions; (3) establish properties of the sum of two reciprocal gamma variables, and related results; (4) study the G distributions (whose Mellin transforms are ratios of products of gamma functions).
"G distributions and the beta-gamma algebra." Electron. J. Probab. 15 2163 - 2199, 2010. https://doi.org/10.1214/EJP.v15-845