Monoidal Alphabets for Generalized Harmonic Sums
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917517988986880 |
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| author | Phadikar, Jayanta |
| author_facet | Phadikar, Jayanta |
| contents | We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21525 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monoidal Alphabets for Generalized Harmonic Sums Phadikar, Jayanta General Mathematics 11M32 (Primary) 05A19, 11M06, 33B30, 68W30 (Secondary) We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities. |
| title | Monoidal Alphabets for Generalized Harmonic Sums |
| topic | General Mathematics 11M32 (Primary) 05A19, 11M06, 33B30, 68W30 (Secondary) |
| url | https://arxiv.org/abs/2605.21525 |