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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)?

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2 Answers 2

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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 $a_{n+1}$.

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  • $\begingroup$ This mens that there are elements whose powers taken as standard part produce Bernoulli numbers, Euler's numbers and whatever other numbers encountered in series for elementary functions. In other words, standard parts of elementary functions of these constants would be expressible as elementary functions of real numbers, in closed form... $\endgroup$
    – Anixx
    Commented Jan 24, 2022 at 23:10
  • $\begingroup$ @Anixx I'm not sure what you mean with your second sentence. What does "standard part of elementary function" mean? And "elementary functions of these constants"? What constants? $\endgroup$
    – Wojowu
    Commented Jan 24, 2022 at 23:14
  • $\begingroup$ See my other answer. Those $p_1$ and $p_2$ would satisfy $\operatorname{st}\frac1{\pi }\ln \left(\frac{p_1-\frac{z}{\pi }}{p_2+\frac{z}{\pi }}\right)=\cot z$, because of this: mathoverflow.net/questions/380142/… $\endgroup$
    – Anixx
    Commented Jan 24, 2022 at 23:20
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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.

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