Abstract

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