  In general, while obtaining the probability density function of sums and
products of shifted random variables, ordinary analytical methods such as
Fourier and Mellin transforms tend to provide integrals which cannot be
expressed in terms of ordinary Meijer G and H functions. This way, the need of
defining new functions which easily enable one to write such integrals in a
closed-form is inherent to the development of this area of statistical
sciences. By generalizing the Mellin transform which defines the H function, a
new function is established. A direct application of the so-called
$\widehat{I}$ is discussed while developing the probability density function of
the sum and the product of shifted generalized gamma random variables.
Important special cases of the $\widehat{I}$ and their applications in science
are also discussed in order to show the applicability of the function hereby
defined.
