  We decouple the Dirac's radial equations in $D+1$ dimensions with
Coulomb-type scalar and vector potentials through appropriate transformations.
We study each of these uncoupled second-order equations in an algebraic way by
using an $su(1,1)$ algebra realization. Based on the theory of irreducible
representations, we find the energy spectrum and the radial eigenfunctions. We
construct the Perelomov coherent states for the Sturmian basis, which is the
basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra.
The physical radial coherent states for our problem are obtained by applying
the inverse original transformations to the Sturmian coherent states.
