Mellin convolutions, statistical distributions and fractional calculus

Abstract

Abstract This paper shows that meaningful interpretations for Mellin convolutions of products and ratios involving two, three or more functions, can be given through statistical distribution theory of products and ratios involving two, three or more real scalar random variables or general multivariate situations. This paper shows that the approach through statistical distributions can also establish connection to fractional integrals, reaction-rate probability integrals in nuclear reaction-rate theory, Krätzel integrals and Krätzel transform in applied analysis, continuous mixtures, Bayesian analysis etc. This paper shows that the theory of Mellin convolutions, currently available for two functions, can be extended to many functions through statistical distributions. As illustrative examples, products and ratios of generalized gamma variables, which lead to Krätzel integrals, reaction-rate probability integrals, inverse Gaussian density etc, and type-1 beta variables, which lead to various types of fractional integrals and fractional calculus in general, are considered.