Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition: $$ A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 + \mathfrak{m}[t^{-1}]) \cdot A^\times \cdot (1 + t A[\![t]\!]) $$ and more concretely $\forall f(t) \in A(\!(t)\!)^\times$: $$ f(t) = t^w \cdot a_0 \cdot \prod_{i=1}^\infty (1-a_{-i} t^{-i}) \cdot \prod_{i=1}^\infty(1-a_i t^i) $$ uniquely, where $w \in \mathbb{Z}$ and $$ a_i \in \begin{cases} 0, \text{ if } i \ll 0, \\ \mathfrak{m}, \text{ if } i < 0, \\ A^\times, \text{ if } i = 0, \\ A, \text{ if } i > 0. \end{cases} $$ The coefficients $a_i$ are apparently called Witt parameters.
This is taken from the 2004 paper Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids over an Artinian Local Ring (arXiv link, DOI link behind paywall) by Anderson and Romo. The factorization is strongly reminiscent of the Weiestrass Factorization Theorem in Complex Analysis (I guess it's its analogue in the formal setting). The authors don't provide a concrete reference for these facts, and I've searched through their bibliography, so I'm guessing this result must be fairly well-known to experts.
In what context do these factorizations appear? I guess I am looking for the right keywords to search for at least, or better yet, for references that explain these identities and maybe related matters. Basically, I'd like to learn more about them. The case of a local Artinian ring is of a special interest to me.
PS: Now that I am aware of the decomposition, it's not difficult to prove it. I am more interested in how it showed up, how one arrived at it as part of what, i.e. its origin story so to speak and context.
