  We prove some interesting multiplicative relations which hold between the
coefficients of the logarithmic derivatives obtained in a few simple ways from
$\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives
connect power sums to elementary symmetric functions via the Newton identities,
we establish, as applications, new identities between important quantities of
function field arithmetic, such as the Bernoulli-Carlitz fractions and power
sums as well as their multi-variable generalizations. Resulting understanding
of their factorizations has arithmetic significance, as well as applications to
function field zeta and multizeta values evaluations and relations between
them. Using specialization/generalization arguments, we provide much more
general identities on linear forms providing a switch between power sums for
positive and negative powers.
