Duality formulas for $\hat{\mathcal{Q}}$-multiple zeta values

Abstract

In this paper, we introduce two duality formulas for $\hat{\mathcal{Q}}$-multiple zeta values ($\hat{\mathcal{Q}}$-MZVs for short) which are defined by Takeyama and Tasaka. Since taking two different limits of $\hat{\mathcal{Q}}$-MZV recovers the $\mathbf{p}$-adic multiple zeta value and the $t$-adic multiple zeta value respectively, finding the relation of $\hat{\mathcal{Q}}$-MZVs partially supports the Kaneko--Zagier conjecture and its refined version. Currently, three types of duality formulas for the $\mathbf{p}$-adic multiple zeta values are known, and for one of them, the $q$-analogue was studied by Takeyama and Tasaka. We present $q$-analogues of the remaining two duality formulas for the $\mathbf{p}$-adic multiple zeta values.

14 pages