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GENERALIZED FINITE POLYLOGARITHMS

Part of: Other special functions Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 February 2020

MARINA AVITABILE  and
SANDRO MATTAREI
MARINA AVITABILE
Affiliation:
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano - Bicocca, via Cozzi 55, I-20125Milano, Italy, e-mail: marina.avitabile@unimib.it
SANDRO MATTAREI
Affiliation:
Charlotte Scott Centre for Algebra, University of Lincoln, Brayford Pool Lincoln, LN6 7TS, United Kingdom, e-mail: smattarei@lincoln.ac.uk
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Abstract

We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ 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 grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

MSC classification

Primary: 33E50: Special functions in characteristic $p$ (gamma functions, etc.)
Secondary: 11G55: Polylogarithms and relations with $K$-theory 39B52: Equations for functions with more general domains and/or ranges 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 66 - 80
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

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