In this paper, we discuss the completely monotonic functions and their relation to some of
the famous special functions such as (Gamma, Kumar, Parabolic cylinder, Gauss
hypergeometric, MacDonald, Whittaker and Generalized Mittag-Leffler) function.
In
addition, the relationship of the completely monotonic integrations with absolute progress
under conditions of convergence such as transformations (Hankel, Lambert, Stieltjes and
Laplace).
We will found other modes of composite functions given in terms of non-negative power
chains and integrative transformations of completely monotonic non-negative functions,
the state of integrative transform functions with a homogeneous nucleus of the first order,
and the logarithmically completely monotonic functions.
The importance of the row of completely monotonic functions that are associated with the
transformation of the Stieltjes defined as a class of special functions regression functions.
Some of the oscillations of these functions resulting from completely monotonic functions
are not decreasing or convex, but most of them are completely monotonic functions.
