Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums

Yoshihiro Takeyama; Yuto Tsuruta

arxiv: 2606.02123 · v1 · pith:CATGG3VFnew · submitted 2026-06-01 · 🧮 math.NT

Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic q-sums

Yoshihiro Takeyama , Yuto Tsuruta This is my paper

Pith reviewed 2026-06-28 12:52 UTC · model grok-4.3

classification 🧮 math.NT

keywords multiple harmonic q-sumsq-analoguesmultiple zeta valuesdualityKawashima functionharmonic sumsq-series identities

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The pith

A q-analogue of the Maesaka-Seki-Watanabe formula for multiple harmonic sums yields an alternative proof of duality for q-multiple zeta values and an identity for the q-Kawashima function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper takes a q-analogue formula previously established for multiple harmonic q-sums and applies it in two directions. One application supplies a fresh route to the known duality relation among q-analogues of multiple zeta values. The other derives an explicit identity that the q-analogue of the Kawashima function must satisfy. A reader cares because these results link a single combinatorial identity to two distinct structures in q-series without invoking the original analytic or generating-function machinery.

Core claim

The authors demonstrate that the q-analogue of the Maesaka-Seki-Watanabe formula, which extends the original identity to Schur-type multiple harmonic q-sums, directly implies both the duality of q-multiple zeta values and a functional identity for the q-Kawashima function.

What carries the argument

The q-analogue formula of Maesaka-Seki-Watanabe type for multiple harmonic q-sums, previously proved by the second author, which is applied term-by-term to the defining series of the two target objects.

If this is right

Duality holds for the q-analogue of multiple zeta values by direct substitution into the q-formula.
The q-Kawashima function satisfies a specific closed identity derived from the same substitution.
Both results follow without extra convergence conditions or auxiliary generating functions.
The method treats the two objects uniformly through the same combinatorial identity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same substitution technique may produce further relations among other q-series that admit Schur-type expansions.
If the base formula extends to higher-depth or colored variants, the duality and Kawashima claims would extend automatically.
The approach suggests that combinatorial sum identities can serve as a uniform engine for q-analogues of classical zeta identities.

Load-bearing premise

The q-analogue formula holds for the specific indices and parameters that appear in the duality and Kawashima settings.

What would settle it

An explicit counterexample to the duality relation for any q-multiple zeta value, or a numerical mismatch in the claimed Kawashima identity at a concrete index and q-value, would falsify the applications.

read the original abstract

Maesaka, Seki and Watanabe proved a formula for multiple harmonic sums. Yamamoto generalized it to Schur-type multiple harmonic sums, and the second author proved a $q$-analogue of this generalization. In this paper, we give two applications of the $q$-analogue formula. The first is an alternative proof of the duality of a $q$-analogue of multiple zeta values. The second is a proof of an identity for a $q$-analogue of the Kawashima function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper applies an existing q-analogue formula to deliver one alternative proof and one new identity in the q-MZV setting.

read the letter

The core contribution is straightforward: the authors take the q-analogue of the Maesaka-Seki-Watanabe formula (already proved by the second author) and apply it twice. One application gives an alternative route to duality for q-analogues of multiple zeta values. The other produces an identity for the q-analogue of the Kawashima function.

The work is clean in its execution. It treats the earlier formula as given and shows direct applicability to the generating functions or sums involved, without adding new restrictions on indices, convergence, or the parameter q. That keeps the derivations short and avoids hidden assumptions.

The limitation is that the paper stays inside a narrow lane. Both results are extensions rather than first-principles advances, and the broader impact on multiple zeta values or harmonic sums is modest. The alternative proof may save some readers time, but it does not reorganize the area or resolve open questions. The new identity is concrete, yet its value hinges on how central the Kawashima function is to ongoing work in this subfield.

This is for people already tracking q-analogues of MZVs and related sums. A specialist in that corner could pick up the alternative proof or the identity without much overhead. The citation pattern is appropriate, the logic holds together, and there are no visible internal contradictions.

I would send it to peer review. It is a modest but honest addition that fits the standards of the niche.

Referee Report

0 major / 2 minor

Summary. The manuscript gives two applications of the q-analogue of the Maesaka-Seki-Watanabe formula for multiple harmonic q-sums (previously established by the second author). The first application supplies an alternative proof of duality for the q-analogue of multiple zeta values. The second derives an identity satisfied by the q-analogue of the Kawashima function. Both applications treat the formula as given and apply it directly to the relevant generating functions or sums.

Significance. If the derivations are correct, the work demonstrates the utility of the q-analogue formula by recovering a known duality result via a different route and by establishing a new identity for the q-Kawashima function. These contributions sit within the existing literature on q-analogues of multiple zeta values and related special functions.

minor comments (2)

The statement of the underlying q-analogue formula (from the second author's earlier work) should be recalled explicitly in the introduction or a preliminary section so that the applications can be followed without external reference.
Notation for the q-multiple zeta values and the q-Kawashima function should be fixed at the first appearance and used consistently; any deviation between the duality statement and the Kawashima identity should be noted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript and recommending minor revision. The report provides a helpful summary of the work but lists no major comments under the MAJOR COMMENTS section. Accordingly, we have no specific points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper takes as given the q-analogue of the Maesaka-Seki-Watanabe formula previously established by the second author and applies it to obtain an alternative proof of duality for q-analogues of multiple zeta values plus an identity for the q-Kawashima function. These applications constitute independent derivations that add new content; they do not reduce by construction to the inputs, involve fitted parameters renamed as predictions, or rely on a self-citation chain that renders the results tautological. The self-citation supplies an external theorem rather than a load-bearing premise internal to this manuscript, leaving the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the q-analogue formula established in the second author's previous work and on standard algebraic properties of q-series and multiple sums.

axioms (2)

domain assumption The q-analogue of the Maesaka-Seki-Watanabe formula holds for the relevant multiple harmonic q-sums.
Invoked as the starting point for both applications; referenced in the abstract as previously proved by the second author.
standard math Standard algebraic identities for q-series and multiple sums remain valid under the q-deformation.
Background assumption required for any manipulation of the q-sums.

pith-pipeline@v0.9.1-grok · 5613 in / 1471 out tokens · 34725 ms · 2026-06-28T12:52:55.540482+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 2 canonical work pages · 1 internal anchor

[1]

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Bradley, D. M., Multipleq-zeta values,J. Algebra283(2005), 752–798

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[2]

and Onozuka, T., Integral expressions for Schur multiple zeta values,Indag

Hirose, M., Murahara, H. and Onozuka, T., Integral expressions for Schur multiple zeta values,Indag. Math.35 (2024), 1197–1211

2024
[3]

and Watanabe, T., Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730

Maesaka, T., Seki, S. and Watanabe, T., Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730

work page arXiv
[4]

Number Theory129(2009), 755–788

Kawashima, G., A class of relations among multiple zeta values,J. Number Theory129(2009), 755–788

2009
[5]

Kawashima, G., Multiple series expressions for the Newton series which interpolate finite multiple harmonic sums, preprint, arXiv:0905.0243

work page internal anchor Pith review Pith/arXiv arXiv
[6]

1, 15–28

Takeyama, Y., Quadratic relations for aq-analogue of multiple zeta values,Ramanujan J.27(2012), no. 1, 15–28

2012
[7]

Takeyama, Y., The Algebra ofq-analogue of multiple harmonic series,Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)9(2013), 15 pages

2013
[8]

Maesaka–Seki–Watanabe’s formula for multiple harmonicq-sums,J

Tsuruta, Y. Maesaka–Seki–Watanabe’s formula for multiple harmonicq-sums,J. Number Theory276(2025), 270–285

2025
[9]

Yamamoto, S., Some remarks on Maesaka–Seki–Watanabe’s formula for the multiple harmonic sums,J. Math. Soc. Japan78No. 1 (2026), 133–148. Department of Mathematics, Institute of Pure and Applied Sciences, University of Tsukuba, Ibaraki 305-8571, Japan Email address:takeyama@math.tsukuba.ac.jp Mathematical Institute, Tohoku University, Sendai 980-8578, Japa...

2026

[1] [1]

M., Multipleq-zeta values,J

Bradley, D. M., Multipleq-zeta values,J. Algebra283(2005), 752–798

2005

[2] [2]

and Onozuka, T., Integral expressions for Schur multiple zeta values,Indag

Hirose, M., Murahara, H. and Onozuka, T., Integral expressions for Schur multiple zeta values,Indag. Math.35 (2024), 1197–1211

2024

[3] [3]

and Watanabe, T., Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730

Maesaka, T., Seki, S. and Watanabe, T., Deriving two dualities simultaneously from a family of identities for multiple harmonic sums, preprint, arXiv:2402.05730

work page arXiv

[4] [4]

Number Theory129(2009), 755–788

Kawashima, G., A class of relations among multiple zeta values,J. Number Theory129(2009), 755–788

2009

[5] [5]

Kawashima, G., Multiple series expressions for the Newton series which interpolate finite multiple harmonic sums, preprint, arXiv:0905.0243

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

1, 15–28

Takeyama, Y., Quadratic relations for aq-analogue of multiple zeta values,Ramanujan J.27(2012), no. 1, 15–28

2012

[7] [7]

Takeyama, Y., The Algebra ofq-analogue of multiple harmonic series,Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)9(2013), 15 pages

2013

[8] [8]

Maesaka–Seki–Watanabe’s formula for multiple harmonicq-sums,J

Tsuruta, Y. Maesaka–Seki–Watanabe’s formula for multiple harmonicq-sums,J. Number Theory276(2025), 270–285

2025

[9] [9]

Yamamoto, S., Some remarks on Maesaka–Seki–Watanabe’s formula for the multiple harmonic sums,J. Math. Soc. Japan78No. 1 (2026), 133–148. Department of Mathematics, Institute of Pure and Applied Sciences, University of Tsukuba, Ibaraki 305-8571, Japan Email address:takeyama@math.tsukuba.ac.jp Mathematical Institute, Tohoku University, Sendai 980-8578, Japa...

2026