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 [O.sub.K], the residue field [F.sub.q], and uniformizing [pi], we consider the Lubin-Tate tower [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [K.sub.n] = K ([[pi].sub.n]), f ([[pi].sub.0]) = 0, and f ([[pi].sub.n+1]) = [[pi].sub.n]. Here n[greater than or equal to]0 f (X) defines the endomorphism [[pi]] of the Lubin-Tate group. If q [not equal to] 2, then for any formal power series g(X) [member of] [O.sub.K] [[X ]] the following equality holds: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. One has a similar equality in the case q = 2. n=0