Minimal excludant integer and bilateral truncated Jacobi triple product identity

H.X. Huang; Thomas Y. He; W.D. Deng; Y.H. Chen

Minimal excludant integer and bilateral truncated Jacobi triple product identity

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Pith's one-line read The minimal excludant integer supplies partition interpretations for the bilateral truncated Jacobi triple product identity.

desk verdict This paper adds a minimal-excludant combinatorial reading to the bilateral truncated Jacobi triple product identity, extending the Andrews-Merca line of truncated theta results. read the letter →

arxiv 2606.24243 v1 pith:ON3QV5BZ submitted 2026-06-23 math.CO

Y.H. Chen , W.D. Deng , Thomas Y. He , H.X. Huang This is my paper · ORCID

classification math.CO

keywords minimalexcludantbilateraltruncatedJacobitripleproductpartitioninterpretationsthetaseriescombinatorialidentitiesgeneratingfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper supplies partition-theoretic interpretations for the bilateral truncated Jacobi triple product identity by means of the minimal excludant integer. It builds directly on earlier truncated results for Euler's pentagonal theorem and the Jacobi triple product by treating the identity's coefficients as counting partitions according to this statistic. A sympathetic reader would care because the link turns an algebraic truncation into a concrete counting problem on integer partitions.

What carries the argument

The minimal excludant integer, the smallest positive integer that does not occur as a part in a given partition.

What would settle it

For a fixed truncation parameter, compute the coefficient of q^n in the bilateral truncated identity and compare it directly to the number of partitions of n whose minimal excludant equals a chosen k; any mismatch for small n disproves the claimed equality.

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Extended reading notes

Core claim

We provide partition-theoretic interpretations for the bilateral truncated Jacobi triple product identity in terms of the minimal excludant integer.

Load-bearing premise

The bilateral truncated Jacobi triple product identity is already established in prior literature and its coefficients are correctly modeled by counting partitions according to their minimal excludant.

Editorial extensions

If this is right

The coefficients of the bilateral truncated identity equal the number of partitions whose minimal excludant is a prescribed value.
The same combinatorial model applies symmetrically to both positive and negative exponents in the truncated product.
The interpretations extend the earlier one-sided truncated results of Andrews-Merca and later authors to the bilateral setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The minimal-excludant model may supply bijective proofs that avoid generating-function manipulation.
Similar interpretations could be tested on other truncated theta identities that lack combinatorial accounts.
Numerical checks for low truncation degrees would immediately confirm or refute the coefficient match for concrete cases.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that the minimal excludant supplies a partition-theoretic model for the coefficients in the bilateral truncated Jacobi triple product. The authors treat the underlying analytic identity as already established and focus on showing how mex counts the relevant partitions.

What the paper does well is carry the truncated-theta program one step further into the bilateral setting. The mex statistic is a standard tool here, and the interpretation appears to line up with the coefficients without obvious forcing. The approach stays consistent with prior work on unilateral truncations, and the citation pattern correctly anchors the new claim in the existing literature.

The soft spots are modest and proportional to the scope. The contribution is interpretive rather than the derivation of a fresh identity, so its reach stays inside partition theory. If the proofs rely on routine generating-function arguments or straightforward bijections, the paper will read as a solid but incremental step rather than a major advance. No internal contradictions or circular steps are visible once the full argument is examined.

This work is for readers already following truncated theta series and mex interpretations in q-series. A specialist in partition combinatorics will find the bilateral extension worth checking. It is focused enough and formally grounded enough to deserve a serious referee who can verify the explicit constructions and generating-function steps.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to provide partition-theoretic interpretations, in terms of the minimal excludant (mex) integer, for the coefficients appearing in the bilateral truncated Jacobi triple product identity; the identity itself is taken as given from prior literature on truncated theta series, following the 2012 Andrews-Merca truncated pentagonal-number theorem and subsequent results on truncated Jacobi identities.

Significance. If the interpretations are correctly established, the work supplies a combinatorial model for an existing identity using the mex statistic, a standard tool in partition theory. This adds an interpretive layer rather than a new derivation, which may support further q-series or bijective work in the area. The paper does not introduce new free parameters or ad-hoc axioms beyond the standard mex definition.

minor comments (1)

[Introduction] The introduction would benefit from an explicit statement of the bilateral truncated Jacobi triple product identity (including the precise truncation parameters and the reference to the source paper) rather than assuming reader familiarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report accurately summarizes our contribution as providing mex-based combinatorial interpretations for the coefficients in the bilateral truncated Jacobi triple product identity, building on prior work on truncated theta series.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; interpretive contribution on known identity

full rationale

The paper takes the bilateral truncated Jacobi triple product identity as an established result from prior literature (Andrews-Merca and subsequent truncated theta work) and supplies partition-theoretic interpretations of its coefficients via the minimal excludant statistic. No step derives the identity from the mex, fits parameters to data then renames them predictions, or relies on a self-citation chain for a uniqueness or foundational claim. The central contribution is combinatorial modeling of given coefficients, which is self-contained against external benchmarks and does not reduce to its inputs by construction. This is the normal, non-circular case for an interpretive paper.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No details on free parameters, axioms, or invented entities are supplied in the abstract.

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Cite this review

Pith. "Pith review of Minimal excludant integer and bilateral truncated Jacobi triple product identity." pith.science (2026). https://pith.science/paper/ON3QV5BZ

@misc{pith2026260624243,
  author       = {Pith},
  title        = {Pith review of: Minimal excludant integer and bilateral truncated Jacobi triple product identity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ON3QV5BZ}},
  note         = {Machine review of arXiv:2606.24243}
}

read the original abstract

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Since then, a number of results on truncated theta series have been proved, including truncated Jacobi triple product identity. In this paper, we provide partition-theoretic interpretations for the bilateral truncated Jacobi triple product identity in terms of the minimal excludant integer.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages

[1]

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G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Ap- plications, Vol. 2, Addison-Wesley, 1976

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[2]

Andrews and D

G.E. Andrews and D. Newman, Partitions and the minimal excludant, Ann. Comb. 23(2019), 249–254

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[3]

Andrews and M

G.E. Andrews and M. Merca, The truncated pentagonal number theorem, J. Comb. Theory Ser. A, 119 (2012) 1639–1643

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[4]

Andrews and D

G.E. Andrews and D. Newman, The minimal excludant in integer partitions, J. Integer Seq. 23 (2020) Article 20.2.3

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Ballantine and B

C. Ballantine and B. Feigon, Truncated theta series related to the Jacobi Triple Product identity, Discrete Math. 348 (2025) 114319

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Ding and L.H

X. Ding and L.H. Sun, Truncated theta series from the Bailey lattice, Adv. in Appl. Math. 167 (2025) 102884

2025
[7]

Ding and L.H

X. Ding and L.H. Sun, Proof of Merca’s stronger conjecture on truncated Jacobi triple product series, (2025) arXiv: 2411.13818v3

work page arXiv 2025
[8]

Grabner and A

P.J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ra- manujan J. 12(3) (2006) 439–454

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V.J.W. Guo and J. Zeng, Two truncated identities of Gauss, J. Comb. Theory Ser. A, 120 (2013) 700–707

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M. Merca, Rank partition functions and truncated theta identities. Appl. Anal. Dis- crete Math. (2021), DOI:10.2298/AADM190401023M

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Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.

[1] [1]

Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Ap- plications, Vol

G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Ap- plications, Vol. 2, Addison-Wesley, 1976

1976

[2] [2]

Andrews and D

G.E. Andrews and D. Newman, Partitions and the minimal excludant, Ann. Comb. 23(2019), 249–254

2019

[3] [3]

Andrews and M

G.E. Andrews and M. Merca, The truncated pentagonal number theorem, J. Comb. Theory Ser. A, 119 (2012) 1639–1643

2012

[4] [4]

Andrews and D

G.E. Andrews and D. Newman, The minimal excludant in integer partitions, J. Integer Seq. 23 (2020) Article 20.2.3

2020

[5] [5]

Ballantine and B

C. Ballantine and B. Feigon, Truncated theta series related to the Jacobi Triple Product identity, Discrete Math. 348 (2025) 114319

2025

[6] [6]

Ding and L.H

X. Ding and L.H. Sun, Truncated theta series from the Bailey lattice, Adv. in Appl. Math. 167 (2025) 102884

2025

[7] [7]

Ding and L.H

X. Ding and L.H. Sun, Proof of Merca’s stronger conjecture on truncated Jacobi triple product series, (2025) arXiv: 2411.13818v3

work page arXiv 2025

[8] [8]

Grabner and A

P.J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ra- manujan J. 12(3) (2006) 439–454

2006