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arXiv · 2601.21395

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

Zikang Dong·Takao Komatsu
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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.

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Zikang Dong, Takao Komatsu. 2026-01-29. Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets. https://arxiv.org/abs/2601.21395

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