Algebraic dilogarithm identities
type: Item
instance of: scholarly article
title: Algebraic dilogarithm identities
zbMATH Open document ID: 0916.11013
DOI: 10.1023/A:1009709927327
author: Basil Gordon
author: Richard J. McIntosh
published in: The Ramanujan Journal
publication date: 1999-07-19T00:00:00Z
review text: The Rogers \(L\)-function \(L(z)= \sum^\infty_{n= 1} x^n/n^2+{1\over 2}\log x\log(1- x)\) has a bewildering array of functional relationships. The evidence is ever more compelling that all are algebraic consequences of Abel's functional equation \[ L(x)+ L(y)= L(xy)+ L\Biggl({x(1- x)\over 1-xy}\Biggr)+ L\Biggl({y(1- x)\over 1-xy}\Biggr). \] The authors derive the simple functional equations of Euler \((L(x)+ L(1- x)= L(1))\) and Landen and a ten term equation of Rogers in this way. It seems Rogers was inclined to believe that his equation could not be derived in this way. The dilogarithm appears in the dominant term of the asymptotic expansion of certain \(q\)-series. Consequently, \(q\)-series identities yield, analytically, functional equations for the dilogarithm. In this way, comparing leading terms of the \(q\)-binomial identity, \(\sum^\infty_{n= 0}(a)_n b^n/(q)_n= (ab)_\infty/(b)_\infty\), gives Abel's functional equation. The authors give a number of examples based on the identities for basic hypergeometric series and show in each case that the resulting dilogarithm identities are obtainable algebraically from Abel's identity. Finally, for certain algebraic numbers \(\theta\), the dilogarithm satisfies a linear relation of the shape \(\sum^n_{k= 1} c_kL(\theta^k)= 0\) with integers \(c_k\) not all \(0\). For example, \(L(1)= \pi^2/6\), \(L(1/2)= \pi^2/12\), \(L(1/9)- 6L(1/3)+ 2L(1)= 0\). These are the only known rational \(\theta\) (but there are other linear relations of this special shape for \(\theta={1\over 2}\) and \({1\over 3}\)). Many examples have been discovered by asymptotic analysis of \(q\)-series and by computer search. Two new examples are cited: \[ 5L(\delta^3)- 5L(\delta)+ L(1)= 0,\text{ where }\delta= (\sqrt{3+ 2\sqrt 5}- 1)/2\;(\delta^4+ 2\delta^3- \delta- 1= 0), \] \[ 2L(a^3)- 2L(a^2)- 11L(a)+ 3L(1)= 0,\text{ where } a= 2\sqrt 3 \cos(5\pi/18)- 2 \] \[ (a^3+ 6a^2+ 3a- 1= 0). \] The authors demonstrate an algebraic proof of the first from Ab…
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 1185665
zbMATH Keywords: Rogers \(L\)-function
zbMATH Keywords: functional equations
zbMATH Keywords: dilogarithm
zbMATH Keywords: \(q\)-series identities
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: Abel's identity
reviewed by: John H. Loxton
MaRDI profile type: Publication
full work available at URL: a:1009709927327
OpenAlex ID: W80481807
description: scientific article; zbMATH DE number 1185665