Proofs of Two Conjectural Identities on Partial Nahm Sums

Changsong Shi; Liuquan Wang

Proofs of Two Conjectural Identities on Partial Nahm Sums

T0 review · 0 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The two conjectural identities (1.11) and (1.12) hold: the remaining rank-two partial Nahm sums are modular infinite products.

desk verdict Clean, detailed proof of the last open Wang–Zeng partial Nahm identities; the first proof has a Maple black box, but the second independent route closes the gap. read the letter →

arxiv 2507.20270 v1 pith:R3ZZA4KC submitted 2025-07-27 math.NT math.CO

Changsong Shi , Liuquan Wang This is my paper · ORCID

classification math.NTmath.CO MSC 05A3011P8433D1533D4511F0311F27

keywords PartialNahmsumsRogers–RamanujantypeidentitiesBaileypairsHecke-typeseriesAppell–Lerchmodularformsq-hypergeometric

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 proves two conjectural identities from the theory of partial Nahm sums, completing the last open family in a recent classification of rank-two examples. The identities say that two double $q$-hypergeometric sums, with denominators $(q;q)_{2i}(q;q)_{2j}$ and $(q;q)_{2i+1}(q;q)_{2j}$, equal explicit modular infinite products. The proof works in two stages: the double sums are first transformed into Hecke-type series using two Bailey pairs at once, and those series are then converted to products by two independent arguments. This matters because modular Nahm sums are expected to be characters of two-dimensional rational conformal field theories, so each new modular case is a new candidate character.

What carries the argument

The central object is a Bailey pair, two sequences $(\alpha_n,\beta_n)$ linked by (2.2), together with the transformation formula (2.10) that converts a double sum of $\beta$'s into a double sum of $\alpha$'s. The key move is to apply that formula with two different Bailey pairs simultaneously: the standard pair (2.3) together with two new pairs (2.4) and (2.5), which are derived inside the paper from established identities. This produces Hecke-type series, meaning indefinite binary quadratic-form sums with a sign factor, which are not directly visible as products. The second half of the machinery reduces those Hecke-type series to $\theta$-products: one route expresses them as Appell–Lerch sums and then applies cancellation identities, and the other route matches them against previously known Hecke-type product identities. The final products are $J_1J_{2,5}$ and $J_1J_{1,5}$, which are modular forms.

What would settle it

Compute, as formal power series in $q$, the coefficient of $q^N$ on both sides of (1.11) and (1.12) for $N=0,1,\ldots,20$; on the left the double sums become finite once the exponent exceeds 20, so this is a finite calculation. Any inequality at any $N$ would refute the theorem. Equally, substituting the two new Bailey pairs (2.4) and (2.5) into the defining relation (2.2) for $n=0,\ldots,20$ checks the step the whole proof relies on.

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Extended reading notes

Core claim

At the center is Theorem 1.1: with $J_{a,m}=j(q^a;q^m)$ and $J_m=(q^m;q^m)_\infty$, the normalized sums $S(q^{1/2})=(q;q)_\infty^2\sum_{i,j\ge0}q^{2ij+i+j}/((q;q)_{2i}(q;q)_{2j})$ and $T(q^{1/2})=(q;q)_\infty^2\sum_{i,j\ge0}q^{2ij+i+3j}/((q;q)_{2i+1}(q;q)_{2j})$ satisfy $S(q^{1/2})=J_1J_{2,5}$ and $T(q^{1/2})=J_1J_{1,5}$. Equivalently, the unnormalized double sums in (1.11) and (1.12) equal $1/((q;q^2)_\infty^2(q^2,q^8;q^{10})_\infty)$ and $1/((q;q^2)_\infty^2(q^4,q^6;q^{10})_\infty)$, respectively. The paper establishes these by first converting the Nahm sums into Hecke-type series of the form $f_{2,3,2}(x,y,q^3)$, then converting those series to modular products.

Load-bearing premise

The proof's load-bearing premise is that the two new Bailey pairs (2.4) and (2.5) are correct; they are derived inside the paper from earlier identities rather than quoted from an independent source, and they supply the input to the transformation that produces the Hecke-type series. A sign or exponent error in either pair would invalidate the subsequent product identities.

Editorial extensions

If this is right

The three rank-two partial Nahm sums attached to the data in (1.10) are modular, since their $q$-series are products of theta functions; this closes the one family left open in the earlier classification.
The left-hand double sums have explicit product forms, so their coefficients can be studied through the modular products, making asymptotics, parity, and congruence questions more accessible.
The two identities are new Rogers–Ramanujan type sum-to-product identities and can be reused in combinatorial interpretations of what the double sums count.
The proof supplies a template for partial Nahm sums whose direct one-Bailey-pair reduction stalls at single sums: pass through Hecke-type series and reduce those to products by either Appell–Lerch sums or a second Hecke identity.

Reading between the lines

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

The two new Bailey pairs (2.4) and (2.5) are not quoted from an independent source, so a direct verification of them to high order is the most focused way to test the proof's first step.
The same two-Bailey-pairs-plus-Hecke-summation route may apply to other rank-two Nahm-type double sums whose direct reduction to single sums does not reach an infinite product; the present proof suggests the Hecke-type stage is a bridge rather than an obstacle.
The appearance of a single modulus, 30, in all the Appell–Lerch reductions hints that the two identities belong to a finite family of similar cancellations; a systematic search over Hecke-type series of the same shape could uncover neighboring identities.
Because modular Nahm sums are expected to be characters of rational conformal field theories, the two products obtained here are candidate characters for specific two-dimensional theories, although the paper does not identify which theories.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things. The two identities are real: the double sums become the predicted modular infinite products, and there are two independent routes to the final step. The first route passes through Appell–Lerch sums and then says "easy and straightforward" with Maple. The second route avoids that black box entirely, using only published Kim–Lovejoy identities and elementary transformations of Hecke-type series. So the result stands even if you are uncomfortable with the first proof's ending.

What is new: the identities were conjectured in Wang–Zeng's paper and are proved here for the first time, completing the last open family in their list of 14. The method is also new for this literature: instead of one Bailey pair, Lemma 2.2 is applied with two Bailey pairs simultaneously, producing Hecke-type double sums rather than theta products directly. That is a legitimate and interesting route. The two new Bailey pairs in Lemma 2.1 are not quoted; they are derived from Slater's identities, and the coefficient extraction works. The author's citation to their own earlier conjecture is natural, not a red flag, and the external machinery from Lovejoy, Hickerson–Mortenson, Kim–Lovejoy, and Slater is used appropriately.

Soft spots, in proportion. Lemma 3.1 is a long algebraic slog. I did not machine-check every line; my low-order expansions on both sides of (3.5) and (3.6) are consistent, so I have no concrete error to point to. The real expository gap is the final step of the first proof: (3.65) to (3.3) and (3.71) to (3.4) are delegated to "the Maple approach in [3]" without showing the output or the underlying theta identity. That is a legitimate complaint, but it is minor rather than serious, because the second proof derives (3.3) and (3.4) from the same Hecke-type series with written-out transformations. I checked the reductions in that second half and they are sound. No constants are fitted anywhere; the products are conclusions, not assumptions.

This paper is for q-series people, especially those tracking Nahm's problem and partial Nahm sums. It completes a small but meaningful family and adds a modularity data point for rational conformal field theory characters. It does not reshape a branch of mathematics, but it is exactly what a serious referee should spend time on. I would send it out and ask the authors to make the Maple step explicit or to state that the second proof is the primary one, then accept.

Referee Report

0 major / 4 minor

Summary. The paper proves two conjectural Rogers–Ramanujan type identities (1.11) and (1.12) proposed by Wang and Zeng for a remaining family of partial Nahm sums. The proof has two stages: first, using a transformation of Lovejoy that combines two Bailey pairs, the double sums S(q) and T(q) are converted into Hecke-type series f_{2,3,2} (Lemma 3.1). Second, two independent finishes are given: one passes through Appell–Lerch sums and a final Maple simplification, and another reduces the Hecke-type series to known Kim–Lovejoy identities (3.72)–(3.73) using only the elementary transformations (2.11), (2.12), (2.14)–(2.16). The new Bailey pairs (2.4) and (2.5) are derived from Slater's identities in Lemma 2.1.

Significance. If the identities are correct, the paper resolves the last open family in Wang and Zeng's investigation of modular partial Nahm sums, establishing modularity of the corresponding Nahm sums. The proof is notable for combining Bailey pairs with Hecke-type series and Appell–Lerch sums, and for offering a second derivation that is fully checkable by hand from published identities. The argument contains no fitted parameters and does not use the target identities as input; the second proof in particular gives a transparent, verifiable route from the double sums to the modular products. This is a concrete and useful contribution to the literature on partial Nahm sums and Rogers–Ramanujan type identities.

minor comments (4)

[§3, First Proof of Theorem 1.1, after Eqs. (3.65) and (3.71)] The final simplifications from (3.65) to (3.3) and from (3.71) to (3.4) are delegated entirely to the Maple routine described in [3]; as written, this makes the first proof not self-contained. Please provide the key simplification steps or a reproducible Maple script, or state explicitly which Frye–Garvan reduction is being invoked, so that a reader can verify the final theta-product simplification without reimplementing the computation.
[Lemma 3.2, Eq. (3.45), and Lemma 3.3, Eq. (3.60)] The displayed products for W2(q) and M8(q) contain the repeated factor J_{8,30} twice in the denominator; while harmless, this is likely a typographical artifact and should be simplified by cancellation or corrected to avoid confusing the reader.
[Section 3, paragraph after Lemma 3.1] There is a typo in the phrase 'Hecek-type series' in the sentence 'The second method is to transform the Hecek-type series'; it should read 'Hecke-type series'.
[Lemma 2.1, Eqs. (2.4) and (2.5)] The derivation of the two new Bailey pairs from Slater's identities is quite compressed; for completeness, please add a few more intermediate steps showing exactly how the rewritten forms (2.7) and (2.9) match the α_n formulas in the Bailey-pair definition (2.2).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identities are derived from the double sums using external q-series machinery and new Bailey pairs derived from Slater; the cited conjecture is the target, not an input.

full rationale

Both proofs of Theorem 1.1 begin from the double sums S(q) and T(q) and derive the modular products, rather than assuming them. Lemma 3.1 transforms the partial Nahm sums into Hecke-type series via Lovejoy's two-Bailey-pair transformation (2.10), and the new Bailey pairs (2.4) and (2.5) are not quoted as black boxes: Lemma 2.1 derives them by rewriting Slater's identities (2.6) and (2.8) and comparing coefficients with the definition of a Bailey pair. The target identities (1.11) and (1.12) are never used as inputs; they appear only as the conclusions. The second proof completes the derivation using the externally cited Kim–Lovejoy product identities (3.72)–(3.73) and the parameter transformations (2.11), (2.12), (2.14), (2.15), with all final reductions (3.75)–(3.82) shown explicitly, so the central claim does not reduce by construction to a fit or to a self-citation. The first proof leaves the last simplification from (3.65) and (3.71) to (3.3) and (3.4) to the Maple approach of Frye–Garvan; this is an exposition/verification gap, not circularity, and the second proof supplies an independent route that avoids that step. The only self-citation of note is to Wang–Zeng [13], whose conjecture is exactly the statement being proved; it is cited as the origin of the problem, not as evidence for the derivation. No fitted parameters, self-definitional equalities, or author-imported uniqueness theorems occur. Accordingly, the paper is self-contained against established external machinery and merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no new postulated entities. The only novel objects are the two Bailey pairs (2.4)-(2.5), which are proven from Slater rather than assumed.

assumptions (5)

standard math Jacobi triple product identity j(z;q) = sum_n (-1)^n q^{n choose 2} z^n
Used as the basic theta-product identity in Section 2 and throughout the Appell-Lerch computations.
domain assumption Bailey pair definition and Lovejoy's transformation formula (2.10)
Quoted from Lovejoy [6, Theorem 1.2] and applied in Lemma 3.1; this is external but standard q-series machinery.
domain assumption Hickerson-Mortenson evaluations for f_{2,3,2} and Appell-Lerch sums (Lemmas 2.3-2.6)
External published results used in the first proof to convert Hecke-type series into Appell-Lerch sums and theta products.
domain assumption Kim-Lovejoy product identities (3.72)-(3.73)
External published identities used in the second proof to match Hecke-type series with the desired modular products.
domain assumption Genericity of Appell-Lerch parameters
Assumed so that no poles occur in the Appell-Lerch sums and theta quotients; this is a standard running hypothesis in the Hickerson-Mortenson framework.

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Pith. "Pith review of Proofs of Two Conjectural Identities on Partial Nahm Sums." pith.science (2026). https://pith.science/paper/R3ZZA4KC

@misc{pith2026250720270,
  author       = {Pith},
  title        = {Pith review of: Proofs of Two Conjectural Identities on Partial Nahm Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3ZZA4KC}},
  note         = {Machine review of arXiv:2507.20270}
}

read the original abstract

Recently, Wang and Zeng investigated modularity of partial Nahm sums and discovered 14 modular families of such sums. They confirmed modularity for 13 families and proposed a conjecture consisting of two Rogers--Ramanujan type identities for the remaining family. We prove these conjectural identities in two steps. First, employing a transformation formula involving two Bailey pairs, we transform the partial Nahm sums into some specific Hecke-type series. Second, using two distinct approaches, we convert these Hecke-type series to the desired modular infinite products.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On a pair of three-colored (mod 10) partition identities
math.CO 2025-09 conditional novelty 6.0 of 10

Two new partition identities are proved: three-colored partitions with certain forbidden differences have generating functions equal to a distinct-parts factor times the first or second Rogers-Ramanujan product.

Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [3]

Frye and F.G

J. Frye and F.G. Garvan, Automatic proof of theta-function identities, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Texts & Monographs in Symbolic Computation (Springer, Cham, 2019), pp. 195–258

work page 2019

[2] [1]

Z. Cao, H. Rosengren and L. Wang, On some double Nahm sums of Zagier, J. Combin. Theory Ser. A. 202 (2024), 105819

work page 2024

[3] [2]

Cherednik and B

I. Cherednik and B. Feigin, Rogers–Ramanujan type identities and Nil-DAHA, Adv. Math. 248 (2013), 1050–1088

2013

[4] [4]

Hickerson and E.T

D.R. Hickerson and E.T. Mortenson, Hecke-type double sums, Appell–Lerch sums and mock theta functions, I, Proc. London Math. Soc. (3) 109 (2014), 382–422

work page 2014

[5] [5]

Kim and J

B. Kim and J. Lovejoy, Ramanujan type partial theta identities and conjugate Bailey pairs, II. Multisums, Ramanujan J. 46 (2018), 743–764

work page 2018

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Lovejoy, Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J

J. Lovejoy, Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J. 29 (2012), no. 1–3, 51–67

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Nahm, Conformal field theory and torsion elements of the Bloch group

W. Nahm, Conformal field theory and torsion elements of the Bloch group. In Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, pp. 67–132. Springer, 2007

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Rogers, Second memoir on the expansion of certain infinite products, Proc

L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318–343