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arXiv 1809.01237math.NT
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广义有限多项式

Generalized finite polylogarithms

Marina Avitabile, Sandro Mattarei

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18 pages
发表信息：
Glasgow Math. J. 63 (2021) 66-80
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摘要：

我们引入了有限多项式$£_{d}^{(0)}(X)=£_d(X)=\sum_{k=1}^{p-1}X^k/k^d$的一个推广$£_{d}^{(α)}(X)$，其特征为$p$，它依赖于一个参数$α$ .特殊情况下$£_{1}^{(α)}(X)$以前研究的逆，在适当的意义上，截断指数的参数化推广，这是有助于在一个{ \em分级开关}技术的非结合代数。在这里，我们将这样的推广到$£_{d}^{(α)}(X)$在一个自然的方式，并研究这些多项式所满足的一些性质。特别是，我们发现多项式$£_{d}^{(α)}(X)$是如何与$£_{1}^{(α)}(X)$的权力，并得出一些后果。

英文摘要：

We introduce a generalization $£_{d}^{(α)}(X)$ of the finite polylogarithms $£_{d}^{(0)}(X)=£_d(X)=\sum_{k=1}^{p-1}X^k/k^d$, in characteristic $p$, which depends on a parameter $α$. The special case $£_{1}^{(α)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a {\em grading switching} technique for non-associative algebras. Here we extend such generalization to $£_{d}^{(α)}(X)$ in a natural manner, and study some properties satisfied by those polynomials. In particular, we find how the polynomials $£_{d}^{(α)}(X)$ are related to the powers of $£_{1}^{(α)}(X)$ and derive some consequences.

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