Algebraic dilogarithm identities
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 Abel's identity. They give two companion relations to the second for the other two roots of the cubic, suggest that all three equations follow from Abel's identity, but do not have a proof. The basis for a systematic attack on such questions is emerging from the work of Zagier and Kirillov.
scientific article; zbMATH DE number 1185665