  In a recent paper [J.-C. Pain, Opt. Spectrosc. ${\bf 218}$, 1105-1109
(2020)], we discussed the link between expectation values of powers of $r$ and
Clebsch-Gordan coefficients. In this short note we provide additional
information, reminding that such a connection is a direct consequence of group
theory. The hydrogenic radial wavefunctions form bases for infinite dimensional
representations of the algebra of the non-compact group $O(2,1)$ and the
expectation values $r^p$ and $r^{-p}$ ($p$ being positive) transform as tensors
with respect to this algebra. As shown a long time ago by Armstrong [L.
Armstrong Jr., J . Phys. (Paris) Suppl. C 4 ${\bf 31}$, 17 (1970)], analysis of
matrix elements of $r^p$ and $r^{-p}$ reveals that the Wigner-Eckart theorem is
valid for this group and that the corresponding Clebsch-Gordan coefficients are
proportional to the usual $SO(3)$ Clebsch-Gordan coefficients. This
proportionality provides simple explanations of the selection rules for
hydrogenic radial matrix elements pointed out by Pasternack and Sternheimer,
and the proportionality of hydrogenic expectation values of $r^p$ and $r^{-p}$
to $3jm$ symbols.
