Generalized finite polylogarithms

TL;DR

This paper introduces a new family of generalized finite polylogarithms in characteristic p, extending previous work and exploring their properties and relations to other algebraic structures.

Contribution

The authors define a new parameterized generalization of finite polylogarithms and analyze their properties, extending prior inverse relationships and applications in non-associative algebra techniques.

Findings

Established relations between generalized polylogarithms and their powers.

Extended the inverse relationship to a broader class of polylogarithms.

Derived properties and structural insights of the new polynomials.

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.