On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields

For a local number field $K$ with the ring of integers $\mathcal O_K$, the residue field $\mathbb F_q$, and uniformizing $\pi$, we consider the Lubin–Tate tower $K_\pi=\bigcup_{n\geq0}K_n$, where $K_n=K(\pi_n)$, $f(\pi_0)=0$, and $f(\pi_{n+1})=\pi_n$. Here $f(X)$ defines the endomorphism $[\pi]$ of the Lubin–Tate group. If $q\neq2$, then for any formal power series $g(X)\in\mathcal O_K[[X]]$ the following equality holds: $\sum_{n=0}^\infty\mathrm{Sp}_{K_n/K}g(\pi_n)=-g(0)$. One has a similar equality in the case $q=2$.