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G distributions and the beta-gamma algebra. (English) Zbl 1226.60021
Summary: 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).

60E05 Probability distributions: general theory
60E07 Infinitely divisible distributions; stable distributions
33B15 Gamma, beta and polygamma functions
44A10 Laplace transform
44A15 Special integral transforms (Legendre, Hilbert, etc.)
60E10 Characteristic functions; other transforms
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