Instances of the product of independent beta random variables and of the Meijer G and Fox H functions with finite representations

The authors show how we may, through the use of some extended product expressions for the Gamma function and some detailed work on the characteristic functions of the negative logarithm of the product of particular independent Beta random variables, whose second parameters are rational, obtain in a single shot (i) simple expressions for both the probability density and cumulative distribution functions of this product, as well as, concomitantly (ii) very simple and easy to compute alternative finite form expressions for instances of the Meijer G and Fox H functions. These alternative expressions are based on the expressions for the probability density and cumulative distribution functions of the Generalized Integer Gamma distribution and they are not used or recognized by the available software.