We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ 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 grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

中文翻译：

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广义有限对数

我们引入一个概括${\rm{\磅}}_d^{(\alpha)}(X)$有限多对数的${\rm{\磅}}_d^{(0)}(X) = {{\rm{\磅}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$, 在特征p，这取决于一个参数α. 特殊情况${\rm{\磅}}_1^{(\alpha)}(X)$作者之前曾将其研究为截断指数的参数化泛化的逆，在适当的意义上，这有助于分级切换非结合代数的技术。在这里，我们将这种概括扩展到${\rm{\磅}}_d^{(\alpha)}(X)$以自然的方式研究这些多项式所满足的一些性质。特别是，我们发现多项式如何${\rm{\磅}}_d^{(\alpha)}(X)$与权力有关${\rm{\磅}}_1^{(\alpha)}(X)$并得出一些后果。