Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of
number fields. In parallel, Pulita has generalised the theory of Dworks power series to a set of power series with coefficients in Lubin-Tate extensions of $Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulitas power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and Lubin-Tate theory, we then generalise Picketts constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other applications are also exposed.
