In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence? 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)?
 A: 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 $a_{n+1}$.
A: It seems, the following elements have the prescribed property (for $B^+(x)$ and $B^-(x)$):
\begin{gather*}
p_0=\varepsilon^{-1} -\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb\\
p_1=\varepsilon^{-1} +\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb 
.
\end{gather*}
The numerators of the terms are given in https://oeis.org/A118050 and the denominators are in https://oeis.org/A118051.
