The parity theorem for multiple polylogarithms
arXiv:1512.04482 · doi:10.1016/j.jnt.2016.08.004
Abstract
We generalize the well-known parity theorem for multiple zeta values (MZV) to functional equations of multiple polylogarithms (MPL). This reproves the parity theorem for MZV with an additional integrality statement, and also provides parity theorems for special values of MPL at roots of unity (also known as coloured MZV). We give explicit formulas in depths 2 and 3 and provide a computer program to compute the functional equations.
21 pages, supplemented by a list functional equations and a Maple program, v2: minor corrections and clarifications, additional references, discussion of singular limits added (from after remark 2.6 to end of section 2), v3: misprint in eq. (3.5) corrected