Relations for Nielsen polylogarithms

Abstract

Polylogarithms appear in many diverse fields of mathematics. Herein, we investigate relations amongst the restricted class of Nielsen-type (essentially, height one) polylogarithms, both generic and at special arguments including the sixth roots of unity. Numerical computations suggest that the collected relations, partially motivated by a previous study of the authors on log-sine integrals, are complete except in the case when the argument is the fundamental sixth root of unity. For use in other applications, all our results are implemented and accessible for use in symbolic computation or to facilitate numeric computation. In particular, the relations are explicitly exhibited in the case of low weights.

The fermion–boson map for large d

“…as a sum of D d (z) functions with odd index. For σ * = 0 the above integral can be easily evaluated using as an intermediate step its representation as a Nielsen generalised polylogarithm S n,p (z) [37,38]…”

“…[36]) that thermal field theories have different analyticity properties with respect to their couplings in d even and odd dimensions. This is also apparent in the calculation of the free energy density of massive free scalars and fermions which for d-odd can be expressed as finite sums of Nielsen's generalized polylogarithms [11,37,38], while for d-even the corresponding expressions are much more complicated. Moreover, we have already noted the relevance of the 1d theories to the physics of the the 3d models.…”

3d fermion-boson map with imaginary chemical potential

“…We find (we give the result for general d) with S d = 2π d/2 /Γ(d/2) the d-dimensional solid angle. The integral above can be evaluated in closed form for d odd, in terms of Nielsen's generalized polylogarithms [33,37] (and eventually as a finite sum of polylogarithms which turn into a Bloch-Wigner-Zagier function after the introduction of a gauge field or an imaginary chemical potential [34].) For d = 3 and m b = 0 we have…”

“…33) where z * = e −βσ * −iβα * ,(3.34) and in the second line we have substituted the gap equations. A few more details on the calculation are given in the Appendix.…”

The fermion - boson map at imaginary chemical potential in odd dimensions

“…[48]) that thermal field theories have different analyticity properties with respect to their couplings in d even and odd dimensions. This is also apparent in the calculation of the free energy density of massive free scalars and fermions which for d-odd can be expressed as finite sums of Nielsen's generalized polylogarithms [11,49,50], while for d-even the corresponding expressions are much more complicated. Moreover, we have already noted the relevance of the 1d theories to the physics of the the 3d models.…”