  The radial expectation values of the probability density of a quantum system
in position and momentum spaces allow one to describe numerous physical
quantities of the system as well as to find generalized Heisenberg-like
uncertainty relations and to bound entropic uncertainty measures. It is known
that the position and momentum expectation values of the main prototype of the
$D$-dimensional Coulomb systems, the $D$-dimensional hydrogenic system, can be
expressed in terms of some generalized hypergeometric functions of the type
$_{p+1}F_p(z)$ evaluated at unity with $p=2$ and $p=3$, respectively. In this
work we determine the position and momentum expectation values in the limit of
large $D$ for all hydrogenic states from ground to very excited (Rydberg) ones
in terms of the spatial dimensionality and the hyperquantum numbers of the
state under consideration. This is done by means of two different approaches to
calculate the leading term of the special functions $_{3}F_2\left(1\right)$ and
$_{5}F_4\left(1\right)$ involved in the large $D$ limit of the position and
momentum quantities. Then, these quantities are used to obtain the generalized
Heisenberg-like and logarithmic uncertainty relations, and some upper and lower
bounds to the entropic uncertainty measures (Shannon, R\'enyi, Tsallis) of the
$D$-dimensional hydrogenic system.
