In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?


Particularly, is there an element $w$ of the field such that the standard part (the zeroth element of the corresponding series) of $w^n$ is $B\_n$ (Bernoulli numbers)?



In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
Yes, it is true, and there even are such elements in the field of formal Laurent series. Specifically, let $a\_n,n\geq 1$ be any sequence of real numbers. We then take a Levi-Civita series
$$z=\varepsilon^{-1}+\sum\_{i=0}^\infty b\_i\varepsilon.$$
We want to show the $b\_i$ can be chosen so that the constant term of $z^n$ is $a\_n$ for $n\geq 1$. The key thing to note is that this constant term will only depend on coefficients $b\_i$ for $i<n$. This lets us define $b\_i$ recursively. Suppose we have constructed $b\_i$ for $i<n$ already. To determine $b\_n$, we consider $z^{n+1}$, and note that the condition imposed by its constant term being $a\_{n+1}$ involves a sum of various combinations of $b\_i,i<n$, but $b\_n$ only appears once, so we can always pick $b\_n$ to make the total sum equal to $a\_{n+1}$.


