A generalization of partition identities of G\"ollnitz-Gordon, Rogers-Ramanujan and Nandi

Motoki Takigiku; Shunsuke Tsuchioka

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A generalization of partition identities of G\"ollnitz-Gordon, Rogers-Ramanujan and Nandi

Motoki Takigiku, Shunsuke Tsuchioka

Representation Theory · Submitted 30 Jun 2026 · 20 references

Pith's verdict: The paper extends Göllnitz-Gordon, Rogers-Ramanujan and Nandi identities to level-2 A^{(2)}_odd modules via new Andrews-Gordon series, with proofs for most cases.

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arXiv ID2606.31821 (version 2)

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CitationMotoki Takigiku, Shunsuke Tsuchioka. "A generalization of partition identities of G\"ollnitz-Gordon, Rogers-Ramanujan and Nandi." arXiv:2606.31821 (2026). https://pith.science/paper/U3MSSIYL

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Editor's note

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The main thing to know is that this paper proposes Andrews-Gordon style series for level 2 standard modules of type A^{(2)}_odd and proves the corresponding sum-product identities except for the A^{(2)}_{6n+3} cases. This directly generalizes the three classical results mentioned.

What is new is the application to the full family of these odd A^{(2)} modules at level 2. The construction takes the known series patterns and scales them to cover more modules than the mod 5, 8 and 14 cases. When the series match the module characters, the sum-product equalities become partition identities in the usual way.

The paper does a clean job of stating the scope and the exceptions up front. It reuses established techniques from the q-series literature without overclaiming novelty in the method itself.

The soft spot is the identification step. The abstract presents the series as proposed for the modules and then proves the algebraic sum-product part. If that equality to the actual characters does not hold, the claimed generalization to the modules falls apart. No cross-check with Weyl-Kac formulas or graded dimension counts is visible here, and the missing cases for 6n+3 leave an incomplete picture. Without the full proofs it is difficult to judge how tightly the algebraic construction is tied to the representation theory.

This work is for people already working on partition identities linked to affine Lie algebra characters. A reader who knows the classical cases and the Andrews-Gordon framework will see the extension quickly.

It deserves peer review. The claims are concrete, the scope is stated clearly, and the result sits inside an established subfield where referees can check the details against existing module character formulas.

Referee report

Major objections (2)

[abstract / main results] The central claim that the proposed series equal the characters of the specified level-2 standard modules is presented only as a proposal (see abstract and the statement of the main results). No comparison with the Weyl-Kac character formula, graded-dimension formulas, or other independent expressions for these characters is supplied, so the step that converts the algebraic identities into statements about module characters remains unverified.
[proof sections] The manuscript asserts that sum-product identities are proved for all cases except A^{(2)}_{6n+3}, yet the supplied text contains no derivation steps, recurrence relations, or verification details for any of the claimed proofs. Without at least an outline of the proof strategy, it is impossible to assess whether the generalizations of the Göllnitz-Gordon, Rogers-Ramanujan, and Nandi identities are correctly established.

Simulated authors' rebuttal

read the rebuttal and 2 point-by-point responses

We thank the referee for their careful reading and comments on our manuscript. We respond point by point to the major comments below.

Referee: [abstract / main results] The central claim that the proposed series equal the characters of the specified level-2 standard modules is presented only as a proposal (see abstract and the statement of the main results). No comparison with the Weyl-Kac character formula, graded-dimension formulas, or other independent expressions for these characters is supplied, so the step that converts the algebraic identities into statements about module characters remains unverified.

Authors: The abstract and main results explicitly use the language of 'propose' for the identification of the q-series with the module characters, generalizing the known cases (Göllnitz-Gordon, Rogers-Ramanujan, Nandi) where equality holds. The manuscript's primary contribution is the proof of the corresponding sum-product identities for these series (except A^{(2)}_{6n+3}). No direct comparison with the Weyl-Kac formula is included because the character equality is presented as a proposal rather than a theorem. We will revise the abstract and introduction to emphasize this distinction and note that the proposal is motivated by pattern matching and initial coefficient agreement with known character formulas. revision: yes
Referee: [proof sections] The manuscript asserts that sum-product identities are proved for all cases except A^{(2)}_{6n+3}, yet the supplied text contains no derivation steps, recurrence relations, or verification details for any of the claimed proofs. Without at least an outline of the proof strategy, it is impossible to assess whether the generalizations of the Göllnitz-Gordon, Rogers-Ramanujan, and Nandi identities are correctly established.

Authors: The submitted version does not contain the detailed derivation steps or an outline of the recurrence relations used in the proofs. The proofs follow the classical approach of deriving recurrences satisfied by the proposed q-series and verifying that the product sides obey the same relations, with initial conditions matching. We will add an outline of this strategy, including the key recurrence relations, to the revised manuscript so that the validity of the generalizations can be assessed. revision: yes

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The paper

The paper claims

Andrews-Gordon type series for level-2 standard modules of type A^{(2)}_{odd} satisfy sum-product identities that generalize Rogers-Ramanujan, Göllnitz-Gordon, and Nandi.

The reading

The paper proposes Andrews-Gordon type series for certain level 2 standard modules of type A^{(2)}_{odd}. It proves the corresponding sum-product identities except for the case of A^{(2)}_{6n+3}. These identities generalize those of Göllnitz-Gordon for modulus 8, Rogers-Ramanujan for modulus 5, and partially Nandi for modulus 14. A sympathetic reader would care because they connect the representation theory of affine Lie algebras to classical partition theory in a new way.

What carries the argument

Andrews-Gordon type series proposed as characters of the modules, which then yield the sum-product partition identities.

Extended reading notes

Core claim

We propose Andrews-Gordon type series for certain level 2 standard modules of type A^{(2)}_{odd}, and prove the corresponding sum-product identities except for A^{(2)}_{6n+3}. These identities generalize the identities of Göllnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).

Load-bearing premise

The proposed Andrews-Gordon type series are exactly the characters of the indicated level-2 standard modules.

Editorial extensions

If this is right

The characters of the indicated modules equal the proposed q-series.
The series admit product expansions that encode restricted partition generating functions.
The identities hold uniformly across the covered families of modules.
The method fails to produce identities for the A^{(2)}_{6n+3} family.

Reading between the lines

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

The pattern of exceptions might point to a structural difference in the highest-weight modules for that arithmetic progression.
Analogous series could be sought for standard modules at other levels or for different affine types.
The new identities might imply previously unknown partition congruences or generating-function relations.

Original abstract

We propose Andrews-Gordon type series for certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$, and prove the corresponding sum-product identities except for $A^{(2)}_{6n+3}$. These identities generalize the identities of G\"ollnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).

Figures

**Figure 1.** The affine Dynkin diagrams A (1) 1 and A (2) 2ℓ−1 1.2. The main result. In this paper, we focus on level 2 of type A (2) 2ℓ−1 (see

Figures reproduced from arXiv:2606.31821 by the authors.

References

Works cited (20)

[1]

A.K. Agarwal, G.E. Andrews and D.M. Bressoud,The Bailey lattice, J.Indian Math.Soc.(N.S.)51(1987), 57–73

View on Pith 1987
[2]

G.E. Andrews,A general theory of identities of the Rogers-Ramanujan type, Bull.Amer.Math.Soc.80(1974), 1033–1052

View on Pith 1974
[3]

K. Bringmann, C. Jennings-Shaffer and K. Mahlburg,Proofs and reductions of various con- jectured partition identities of Kanade and Russell, J.Reine Angew.Math.766(2020), 109– 135

View on Pith 2020
[4]

H.-C. Chan,An invitation to q-series. From Jacobi’s triple product identity to Ramanujan’s “most beautiful identity”, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2011)

View on Pith 2011
[5]

K. Ito,Level 2 standard modules forA (2) 9 and partition conditions of Kanade-Russell, J.Algebra667(2025), 746–777

View on Pith 2025
[6]

S. Kanade and M.C. Russell,Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, Electron.J.Combin.26(2019), 33pp

View on Pith 2019
[7]

J. Lepowsky and R. Wilson,The structure of standard modules. I. Universal algebras and the Rogers-Ramanujan identities, Invent.Math.77(1984), 199–290

View on Pith 1984
[8]

J. Lepowsky and R. Wilson,The structure of standard modules. II. The caseA (1) 1 , principal gradation, Invent.Math.79(1985), 417–442

View on Pith 1985
[9]

J. Mc Laughlin, Topics and Methods inq-series. Monographs in Number Theory, 8. World Scientific Publishing Co.Pte.Ltd., Hackensack, NJ, 2018

View on Pith 2018
[10]

D. Nandi,Partition identities arising from the standardA (2) 2 -modules of level 4, Ph.D. Thesis (Rutgers University), 2014

View on Pith 2014
[11]

H. Rosengren,Proofs of some partition identities conjectured by Kanade and Russell, Ra- manujan J.61(2023), 295–317

View on Pith 2023
[12]

A.V. Sills,An invitation to the Rogers-Ramanujan identities. With a foreword by George E. Andrews, CRC Press, (2018)

View on Pith 2018
[13]

S. Tsuchioka,A vertex operator reformulation of the Kanade-Russell conjecture modulo 9, Ramanujan J.65(2024), 217–240

View on Pith 2024
[14]

S. Tsuchioka,An example ofA 2 Rogers-Ramanujan bipartition identities of level 3, J.Lond.Math.Soc. (2)111(2025), no.4, Paper No.e70152

View on Pith 2025
[15]

M. Takigiku and S. Tsuchioka,A proof of conjectured partition identities of Nandi, Amer.J.Math.146(2024) 405–433

View on Pith 2024
[16]

M. Takigiku and S. Tsuchioka,Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebraA (2) 2 , Proc.Amer.Math.Soc.149(2021) 2763–2776

View on Pith 2021
[17]

S.O. Warnaar,50 years of Bailey’s lemma, Algebraic combinatorics and applications (G¨ oßweinstein, 1999), 333–347, Springer, Berlin, 2001

View on Pith 1999
[18]

S.O. Warnaar,q-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem, Ramanujan J.8(2004), 467–474

View on Pith 2004
[19]

H. Wilf and D. Zeilberger,An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities, Invent.Math.108(1992), 575–633

View on Pith 1992
[20]

H. Wilf and D. Zeilberger,Rational functions certify combinatorial identities, J.Amer.Math.Soc.3(1990), 147–158. Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan (current affiliation : Dharmacapital Pte. Ltd.) Department of Mathematical and Computing Sciences, Institute of Science Tokyo, Tokyo 152-8550, Japan ...

View on Pith 1990

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A.K. Agarwal, G.E. Andrews and D.M. Bressoud,The Bailey lattice, J.Indian Math.Soc.(N.S.)51(1987), 57–73

View on Pith 1987

[2] [2]

G.E. Andrews,A general theory of identities of the Rogers-Ramanujan type, Bull.Amer.Math.Soc.80(1974), 1033–1052

View on Pith 1974

[3] [3]

K. Bringmann, C. Jennings-Shaffer and K. Mahlburg,Proofs and reductions of various con- jectured partition identities of Kanade and Russell, J.Reine Angew.Math.766(2020), 109– 135

View on Pith 2020

[4] [4]

H.-C. Chan,An invitation to q-series. From Jacobi’s triple product identity to Ramanujan’s “most beautiful identity”, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2011)

View on Pith 2011

[5] [5]

K. Ito,Level 2 standard modules forA (2) 9 and partition conditions of Kanade-Russell, J.Algebra667(2025), 746–777

View on Pith 2025

[6] [6]

S. Kanade and M.C. Russell,Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, Electron.J.Combin.26(2019), 33pp

View on Pith 2019

[7] [7]

J. Lepowsky and R. Wilson,The structure of standard modules. I. Universal algebras and the Rogers-Ramanujan identities, Invent.Math.77(1984), 199–290

View on Pith 1984

[8] [8]

J. Lepowsky and R. Wilson,The structure of standard modules. II. The caseA (1) 1 , principal gradation, Invent.Math.79(1985), 417–442

View on Pith 1985

[9] [9]

J. Mc Laughlin, Topics and Methods inq-series. Monographs in Number Theory, 8. World Scientific Publishing Co.Pte.Ltd., Hackensack, NJ, 2018

View on Pith 2018

[10] [10]

D. Nandi,Partition identities arising from the standardA (2) 2 -modules of level 4, Ph.D. Thesis (Rutgers University), 2014

View on Pith 2014

[11] [11]

H. Rosengren,Proofs of some partition identities conjectured by Kanade and Russell, Ra- manujan J.61(2023), 295–317

View on Pith 2023

[12] [12]

A.V. Sills,An invitation to the Rogers-Ramanujan identities. With a foreword by George E. Andrews, CRC Press, (2018)

View on Pith 2018

[13] [13]

S. Tsuchioka,A vertex operator reformulation of the Kanade-Russell conjecture modulo 9, Ramanujan J.65(2024), 217–240

View on Pith 2024

[14] [14]

S. Tsuchioka,An example ofA 2 Rogers-Ramanujan bipartition identities of level 3, J.Lond.Math.Soc. (2)111(2025), no.4, Paper No.e70152

View on Pith 2025

[15] [15]

M. Takigiku and S. Tsuchioka,A proof of conjectured partition identities of Nandi, Amer.J.Math.146(2024) 405–433

View on Pith 2024

[16] [16]

M. Takigiku and S. Tsuchioka,Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebraA (2) 2 , Proc.Amer.Math.Soc.149(2021) 2763–2776

View on Pith 2021

[17] [17]

S.O. Warnaar,50 years of Bailey’s lemma, Algebraic combinatorics and applications (G¨ oßweinstein, 1999), 333–347, Springer, Berlin, 2001

View on Pith 1999

[18] [18]

S.O. Warnaar,q-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem, Ramanujan J.8(2004), 467–474

View on Pith 2004

[19] [19]

H. Wilf and D. Zeilberger,An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities, Invent.Math.108(1992), 575–633

View on Pith 1992

[20] [20]

H. Wilf and D. Zeilberger,Rational functions certify combinatorial identities, J.Amer.Math.Soc.3(1990), 147–158. Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan (current affiliation : Dharmacapital Pte. Ltd.) Department of Mathematical and Computing Sciences, Institute of Science Tokyo, Tokyo 152-8550, Japan ...

View on Pith 1990