Parity result for q- and elliptic analogues of multiple polylogarithms

Abstract

It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions Lk(a, α; p, q) , which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result.