Supernumary polylogarithmic ladders and related functional equations
type: Item
instance of: scholarly article
title: Supernumary polylogarithmic ladders and related functional equations
zbMATH Open document ID: 0729.33015
DOI: 10.1007/BF01833152
author: L. Lewin
author: M. D. Abouzahra
published in: Aequationes Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137353
review text: This paper is one of a series in which the authors explore relations between special values of the polylogarithm \(Li_ n(z)=\sum^{\infty}_{k=1}z^ k/k^ n\). The development is now exposed as a whole in the monograph ``Structural properties of polylogarithms'' edited by the first author [Mathematical Surveys and Monographs. 37. Providence, RI: American Mathematical Society (AMS) (1991; Zbl 0745.33009)]. Set \[ L_ n(N,u)=\frac{Li_ n(u^ N)}{N^{n-1}}-\left\{\sum_{r}\frac{A_ rLi_ n(u^ r)}{r^{n-1}}+\frac{A_ 0\log^ n u}{n!}\right\} \] and \[ L_ n=\sum^{n}_{m=2}\frac{B_ m\zeta (m)\log^{n-m} u}{(n-m)!}. \] A ladder is a relation \(L_ n(N,u)=L_ n\) with rational \(A_ r\), \(B_ m\). In all cases found so far, the \(A_ r\) are determined by the cyclotomic equation for the base \(u\): \[ 1-u^ N=u^{-A_ 0}\prod_{r}(1-u^ r)^{A_ r}. \] For \(n\leq 5\), Kummer's functional equations for \(Li_ n\) give many ladders, but there are no functional equations of this type for \(n\geq 6\). For certain special bases, there are additional relations for the dilogarithm which in some cases have analytic proofs and in some cases are still only verified numerically. These can be used to eliminate unwanted transcendental terms from the ladders and so obtain ladders for \(n\geq 6\). Several examples are given for bases satisfying equations of the shape \(u^ p+u^ q=1\). The examples have led the authors to discover some new one variable functional equations for polylogarithms, but these are too complex to reproduce here. These successes and the elimination of other potentialities rely on extensive computer algebra and MACSYMA.
Mathematics Subject Classification ID: 33B30
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 4204809
zbMATH Keywords: special values of polylogarithm
zbMATH Keywords: ladders
zbMATH Keywords: new one variable functional equations for polylogarithms
reviewed by: John H. Loxton
describes a project that uses: MACSYMA
MaRDI profile type: Publication
cites work: Q4744559
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OpenAlex ID: W4253558274
description: scientific article; zbMATH DE number 4204809