  In the research literature, one can find distinct notions for higher order
averaged functions of regularly perturbed non-autonomous $T$- periodic
differential equations of the kind $x'=\varepsilon F(t,x,\varepsilon)$. By one
hand, the classical (stroboscopic) averaging method provides asymptotic
estimates for its solutions in terms of some uniquely defined functions
$\mathbf{g}_i$'s, called averaged functions, which are obtained through
near-identity stroboscopic transformations and by solving homological
equations. On the other hand, a Melnikov procedure is employed to obtain
bifurcation functions $\mathbf{f}_i$'s which controls in some sense the
existence of isolated $T$-periodic solutions of the differential equation
above. In the research literature, the bifurcation functions $\mathbf{f}_i$'s
are sometimes likewise called averaged functions, nevertheless, they also
receive the name of Poincar\'{e}-Pontryagin-Melnikov functions or just Melnikov
functions. While it is known that $\mathbf{f}_1=T \mathbf{g}_1,$ a general
relationship between $\mathbf{g}_i$ and $\mathbf{f}_i$ is not known so far for
$i\geq 2.$ Here, such a general relationship between these two distinct notions
of averaged functions is provided, which allows the computation of the
stroboscopic averaged functions of any order avoiding the necessity of dealing
with near-identity transformations and homological equations. In addition, an
Appendix is provided with implemented Mathematica algorithms for computing both
higher order averaging functions.
