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

Multiplicative relations between coefficients of logarithmic derivatives of $\mathbb{F}_q$-linear functions and applications