Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets

Abstract

Closed-form results for finite $q$-multiple harmonic sums are abundant for uniformly repeated indices, whereas mixed-weight tuples pose substantial combinatorial challenges. Building on recent progress for patterns containing a single weight-$2$ entry embedded among weight-$1$ indices, we treat multisets composed of arbitrary numbers of $1$s and $2$s at roots of unity. Since individual ordered sums resist simple evaluation, we analyze their symmetrized sum over all permutations. Using newly derived evaluations for $q$-multiple sums with negative powers together with multiplicative identities, we obtain compact binomial-based closed forms. These enlarge the repertoire of explicitly computable finite $q$-multiple zeta-type objects and lay groundwork for further mixed-index investigations.