Generalized finite polylogarithms

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.

18 pages

References in corpus (1)

• Exponential functions in prime characteristic