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

In this paper, we introduce two duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values ($\widehat{\mathcal{Q}}$-MZVs for short) which are defined by Takeyama and Tasaka. Since taking two different limits of $\widehat{\mathcal{Q}}$-MZV recovers the $\mathit{p}$-adic multiple zeta value and the $t$-adic multiple zeta value respectively, finding the relation of $\widehat{\mathcal{Q}}$-MZVs partially supports the Kaneko–Zagier conjecture and its refined version. Currently, three types of duality formulas for the $\mathit{p}$-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 $\mathit{p}$-adic multiple zeta values.