  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.
