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

[2601.21395] Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2601.21395 (math) [Submitted on 29 Jan 2026 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets Authors: Zikang Dong , Takao Komatsu View PDF HTML (experimental) 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. Subjects: Number Theory (math.NT) Cite as: arXiv:2601.21395 [math.NT] (or arXiv:2601.21395v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2601.21395 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Takao Komatsu [ view email ] [v1] Thu, 29 Jan 2026 08:32:44 UTC (11 KB) [v2] Tue, 25 Aug 2026 08:23:20 UTC (12 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-01 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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