# Generalized finite polylogarithms

Avitabile, Marina and Mattarei, Sandro (2020) Generalized finite polylogarithms. Glasgow Mathematical Journal, 63 (1). pp. 66-80. ISSN 0017-0895

Full content URL: https://doi.org/10.1017/S0017089520000026

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## 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.

Keywords: finite polylogarithm, Laguerre polynomial, functional equation G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science 40146 06 Mar 2020 08:21

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