Supernumary polylogarithmic ladders and related functional equations
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.
new one variable functional equations for polylogarithms
scientific article; zbMATH DE number 4204809