Generalized finite polylogarithms

We introduce a generalization of the finite polylogarithms, in characteristic p, which depends on a parameter α. The special case 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 in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials are related to the powers of and derive some consequences.