Finite $q$-multiple harmonic sums on $2-\cdots -2,1-\cdots -1$ indices

There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots -A$ indices, that is, the indices are the same. Though the way to treat $q$-multiple zeta values unless the indices are the same, it has been successful to get the explicit expression of $q$-harmonic sums on $1-\cdots -1,2,1-\cdots -1$ indices. In this paper, we shall consider more general results when the ratio of indices of $2$ to indices of $1$ increases.