Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums

Maesaka, Seki and Watanabe proved a formula for multiple harmonic sums. Yamamoto generalized it to Schur-type multiple harmonic sums, and the second author proved a $q$-analogue of this generalization. In this paper, we give two applications of the $q$-analogue formula. The first is an alternative proof of the duality of a $q$-analogue of multiple zeta values. The second is a proof of an identity for a $q$-analogue of the Kawashima function.