  It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler,
$\alpha$-divergences, and Jeffreys' divergences between densities belonging to
a same exponential family have generic closed-form formulas relying on the
strictly convex and real-analytic cumulant function characterizing the
exponential family. In this work, we report (dis)similarity formulas which
bypass the explicit use of the cumulant function and highlight the role of
quasi-arithmetic means and their multivariate mean operator extensions. In
practice, these cumulant-free formulas are handy when implementing these
(dis)similarities using legacy Application Programming Interfaces (APIs) since
our method requires only to partially factorize the densities canonically of
the considered exponential family.
