  The $\eta^\prime$ transition form factor is reanalyzed in view of the recent
BESIII first observation of the Dalitz decay $\eta^\prime\to\gamma e^+e^-$ in
both space- and time-like regions at low and intermediate energies using the
Pad\'e approximants method. The present analysis provides a suitable
parameterization for reproducing the measured form factor in the whole energy
region and allows to extract the corresponding low-energy parameters together
with a prediction of its values at the origin, related to
$\Gamma_{\eta^\prime\to\gamma\gamma}$, and the asymptotic limit. The
$\eta$-$\eta^\prime$ mixing is reassessed within a mixing scheme compatible
with the large-$N_c$ chiral perturbation theory at next-to-leading order, with
particular attention to the OZI-rule--violating parameters. The $J/\psi$,
$Z\to\eta^{(\prime)}\gamma$ decays are also considered and predictions
reported.
