Generalized finite polylogarithms

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.