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

Abstract

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices. Though there is 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 $1-\cdots-1,A,1-\cdots-1$ indices when $A=2$. In this paper, we shall consider more general results when $A\ge 3$.