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

TL;DR

This paper investigates explicit formulas for finite $q$-multiple harmonic sums with indices composed of blocks of 2's and 1's, extending previous results to more general index ratios.

Contribution

It introduces new explicit expressions for $q$-harmonic sums with mixed $2$ and $1$ indices, expanding the understanding of their structure beyond equal or simple index patterns.

Findings

Derived explicit formulas for sums with increasing ratios of 2's to 1's

Extended known results to more complex index structures

Provided new insights into $q$-multiple harmonic sums

Abstract

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.