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The radial expectation values of the probability density of a quantum systemin position and momentum spaces allow one to describe numerous physicalquantities of the system as well as to find generalized Heisenberg-likeuncertainty relations and to bound entropic uncertainty measures. It is knownthat the position and momentum expectation values of the main prototype of the$D$-dimensional Coulomb systems, the $D$-dimensional hydrogenic system, can beexpressed 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 thiswork we determine the position and momentum expectation values in the limit oflarge $D$ for all hydrogenic states from ground to very excited (Rydberg) onesin terms of the spatial dimensionality and the hyperquantum numbers of thestate under consideration. This is done by means of two different approaches tocalculate 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 andmomentum quantities. Then, these quantities are used to obtain the generalizedHeisenberg-like and logarithmic uncertainty relations, and some upper and lowerbounds to the entropic uncertainty measures (Shannon, R\'enyi, Tsallis) of the$D$-dimensional hydrogenic system.

Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D