Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity

Abstract

The Kaneko–Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous works with H. Bachmann and Y. Takeyama, we proved that the finite and symmetric multiple zeta values are obtained as an ‘algebraic’ and ‘analytic’ limit at $$q\rightarrow 1$$ of certain multiple harmonic q-sums, and studied their relations in order to give partial evidence of the Kaneko–Zagier conjecture. In this paper, we start with multiple harmonic q-sums of level N, which are q-analogues of the truncated colored multiple zeta values. We introduce our finite and symmetric colored multiple zeta values as an algebraic and analytic limit of the multiple harmonic q-sums of level N and discuss a higher level (or a cyclotomic) analogue of the Kaneko–Zagier conjecture.