REVIEW 2 major objections 4 minor 8 references
Some results on vanishing coefficients in infinite product expansions
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The coefficient sequences of four q-products are residue-class shifts, so each vanishing theorem reduces to one check.
desk verdict Theorems 1.8 and 1.9 are solid, the self-cited Lemma 1.17 spot-checks correctly, and the main argument holds; the unproved Theorems 1.12–1.15 and the asserted cancellations in Theorem 1.10 are real completeness gaps that keep this at conditional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument is carried by the Rogers-Ramanujan-type quotient $R(q)=(q,q^4;q^5)_\infty/(q^2,q^3;q^5)_\infty$ and by two identities for it quoted from the authors' earlier paper [3]: one expressing $1/(R(q)R(q^2)^2)-q^2R(q)R(q^2)^2$ and the other expressing $R(q^2)/R(q)^2-R(q)^2/R(q^2)$ as products of $\theta$ functions. These identities convert the difference of the two generating functions in each pair into a single explicit infinite product; Jacobi's triple product identity then reads off the coefficients residue class by residue class modulo 5. Theta-function identities from [4] supply the elementary transformations that prepare the products for that comparison.
What would settle it
Expand the four products to a fixed order, say through $q^{20}$, and compare the asserted shifted coefficients, for instance checking $b_2=a_0$, $b_3=a_1$, $b_4=a_2$, $c_5=d_5$, $c_7=d_7$, and the positivity of $c_1-d_1$; any mismatch is a direct disproof. The sharper check is symbolic: the difference of the $a$- and $b$-generating functions must equal $(q^5;q^5)_\infty^4/(q^{10};q^{10})_\infty^4$, and the $c$-$d$ difference must equal $4q(q^{10};q^{10})_\infty^4/(q^5;q^5)_\infty^4$, so the first nonzero coefficient of either error term settles the claim.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is a set of coefficient-identity equivalences. With $\sum_{n=0}^{\infty}a_nq^n=(-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$, $\sum_{n=0}^{\infty}b_nq^n=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, and the complementary definitions for $c_n$ and $d_n$, the paper proves the nine identities listed in the abstract. It also proves explicit difference formulas that pin down the one non-equal residue class: $\sum_{n=0}^{\infty}b_{5n}q^n-\sum_{n=1}^{\infty}a_{5n-2}q^n=(q^5;q^5)_\infty^4/(q^{10};q^{10})_\infty^4$ and $\sum_{n=0}^{\infty}c_{5n+1}q^n-\sum_{n=0}^{\infty}d_{5n+1}q^n=4(q^2;q^2)_\infty^4/(q;q)_\infty^4$. Because these differences are explicit products, extracting the $q^{5n+r}$ terms gives the equalities, and the second difference explains why $c_{5n+1}>d_{5n+1}$ with no other inequality among the residue classes. The same elementary technique also proves several new vanishings for nearby products, recorded in Theorems 1.10, 1.12, 1.13, 1.14, and 1.15.
Load-bearing premise
The argument depends on two unproved formulas quoted from an earlier paper; if either formula is misstated or false, the difference identities and therefore the coefficient equalities would fail.
Editorial extensions
If this is right
- Hirschhorn's two vanishing statements become dependent: from $b_{5n+1}=a_{5n-1}$ and $b_{5n+4}=a_{5n+2}$, knowing either vanishing theorem gives the other.
- Tang's two vanishing statements are linked by the equalities $c_{5n+3}=d_{5n+3}$, $c_{5n+4}=d_{5n+4}$, $c_{5n}=d_{5n}$, $c_{5n+2}=d_{5n+2}$, and the only non-identical residue class, $1 \bmod 5$, is governed by $c_{5n+1}-d_{5n+1}=4(q^2;q^2)_\infty^4/(q;q)_\infty^4$.
- The proofs use only the Jacobi triple product identity and elementary manipulations, so the results bypass the more specialized summation machinery used in earlier vanishing-coefficient theorems.
- The same method yields new vanishings for sign variants, as in Theorem 1.10, and for eight further products in Theorems 1.12 through 1.15, including cases Tang did not list.
Reading between the lines
- Inference: the same residue-shift pattern should occur for other prime moduli; for moduli $p=3$ or $7$, one can define paired products with $p$-step shifts and ask whether the analogous $R(q)$ quotient satisfies a two-term identity of the same shape.
- Inference: the strict inequality $c_{5n+1}>d_{5n+1}$ has a possible combinatorial reading, since the difference generating function $4(q^2;q^2)_\infty^4/(q;q)_\infty^4$ counts colored partitions; the authors do not pursue that interpretation.
- Inference: an independent proof of the two quoted $R(q)$ identities, or a check that they follow from the same theta-function lemmas, would remove the paper's main unverified dependency.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies four q-series whose coefficients are denoted a_n, b_n, c_n, and d_n, all arising from products related to the Rogers-Ramanujan continued fraction. The main results, Theorems 1.8 and 1.9, state that in five residue classes modulo 5 the coefficients satisfy a_{5n}=b_{5n+2}, a_{5n+1}=b_{5n+3}, a_{5n+2}=b_{5n+4}, a_{5n-1}=b_{5n+1}, and c_{5n}=d_{5n}, c_{5n+2}=d_{5n+2}, c_{5n+3}=d_{5n+3}, c_{5n+4}=d_{5n+4}, with c_{5n+1}>d_{5n+1}. These equalities imply that the vanishing theorems of Hirschhorn and Tang each reduce to a single residue-class check. The derivations in Sections 2 and 3 rewrite the products in terms of a quotient R(q) and then apply two identities quoted from the authors' earlier paper [3]. The paper also states further vanishing results in Theorems 1.10 and 1.12-1.15, with Theorem 1.10 proved only in compressed form and Theorems 1.12-1.15 left unproved.
Significance. If the coefficient equalities in Theorems 1.8 and 1.9 hold, they give a clean structural explanation of the previously independent vanishing results of Hirschhorn and Tang: the two vanishing statements in each pair are not independent but forced by a single residue class plus the equalities. The proof strategy is elementary and transparent, and the algebra in Sections 2 and 3 is coherent; conditional on Lemma 1.17, those two theorems are proved in full. The additional results, however, are not supported to the same standard: Theorem 1.10 contains an asserted cancellation with the details omitted, and Theorems 1.12-1.15 are stated without any proof. These gaps are substantial because those results are advertised as part of the paper's contribution, so the manuscript is not yet complete in its current form.
major comments (2)
- [Section 4, proof of Theorem 1.10] The proof asserts, without derivation, that 'it can be shown' the 3-components of S_1,...,S_8 are the eight displayed double sums and that the 4-components of T_1,...,T_8 are the corresponding eight double sums. These cancellations are the entire content of the theorem, and the reader is given no indication of the change of variables or the way the pairs cancel. Please supply the missing extraction, or at least a detailed and checkable outline, for both the e_n and f_n cases.
- [Section 1, Theorems 1.12-1.15] Theorems 1.12-1.15 are stated as results but no proofs are provided; the text says only that 'Since the proofs of Theorems 1.12-1.15 are similar in nature, we omit the proofs.' As written, these are unsupported claims. The abstract advertises 'some other comparable results not listed by Tang,' so these theorems are part of the paper's contribution. They should either be proved, moved to a remark or conjecture, or explicitly identified as quoted results with citations.
minor comments (4)
- [Eqs. (1.5) and (1.14)] The symbols f_1 and f_2 are used without definition. Define f_k := (q^k; q^k)_\infty at first use.
- [Section 4, definition of U_3] The notation '(±q^4; ±q^6; q^{10})_\infty' is ambiguous; it should be written as '(±q^4, ±q^6; q^{10})_\infty', consistent with the product notation used elsewhere.
- [Lemma 1.17] The central difference formulas (1.5) and (1.14) depend on the two identities quoted from [3, Eqs. (1.19)-(1.20)], but those equations are not restated here. Since these identities are the only non-elementary input to Theorems 1.8 and 1.9, please restate them explicitly or include a short proof sketch so that the main argument is verifiable without consulting [3].
- [Throughout] There are typographical errors: 'Ramanuajn' should be 'Ramanujan' in Section 1, and 'diving' should be 'dividing' in the proof of Theorem 1.9. The sign bookkeeping in Section 4 would also be much easier to follow if the upper-sign and lower-sign cases were treated separately.
Circularity Check
No circularity: the coefficient equalities are derived from independent q-series identities, not from the claims themselves.
full rationale
The paper's central results, Theorems 1.8 and 1.9, are derived by rewriting the four q-products in terms of the Rogers-Ramanujan continued fraction R(q) and then applying Lemma 1.17. The lemma is quoted from the authors' earlier paper [3], but it is an auxiliary product identity, not equivalent to the target coefficient equalities. It is parameter-free, explicitly stated, and its assumptions do not include any of the conclusion coefficients; hence it constitutes independent support rather than a circular input. In Section 2, the proof computes B(q) - q^2 A(q) as a prefactor times (1/(R(q)R(q^2)^2) - q^2 R(q)R(q^2)^2), and Lemma 1.17 converts this to (q^5;q^5)^4/(q^{10};q^{10})^4, a series supported only on exponents divisible by 5. Coefficient extraction then forces the equalities in (1.6)-(1.9) directly. Similarly, Section 3 reduces C(q)-D(q) to 4q (q^{10};q^{10})^4/(q^5;q^5)^4, yielding (1.10)-(1.14). No parameter is fitted, no coefficient is defined in terms of a conclusion, and no uniqueness theorem is invoked to make a choice forced. The paper does omit proofs of Theorems 1.12-1.15 and abbreviates the cancellation in Section 4 as 'Proceeding as in [5]', but these are completeness and verification gaps, not circularity. The self-citation to [3] is load-bearing in the proof but does not make the derivation equivalent to its inputs, because the quoted identities are checkable independent facts that do not encode the vanishing or equality claims being proved. Accordingly, no circular step is identified.
Assumptions & free parameters
assumptions (3)
- standard math Jacobi triple product identity f(a,b) = (-a;ab)_∞(-b;ab)_∞(ab;ab)_∞
- standard math Ramanujan's theta function identities in Lemma 1.16 (from Berndt [4])
- standard math Two identities for R(q) = (q,q^4;q^5)_∞/(q^2,q^3;q^5)_∞ in Lemma 1.17, cited from Baruah and Begum [3]
Cite this review
Pith. "Pith review of Some results on vanishing coefficients in infinite product expansions." pith.science (2026). https://pith.science/paper/XJAWFOUM
@misc{pith2026190807737,
author = {Pith},
title = {Pith review of: Some results on vanishing coefficients in infinite product expansions},
year = {2026},
howpublished = {\url{https://pith.science/paper/XJAWFOUM}},
note = {Machine review of arXiv:1908.07737}
}
abstract
Recently, M. D. Hirschhorn proved that, if $\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$ and $\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, D. Tang proved some comparable results including the following: If $ \sum_{n=0}^\infty c_nq^n := (-q,-q^4;q^5)_\infty^3(q^3,q^7;q^{10})_\infty$ and $\sum_{n=0}^\infty d_nq^n := (-q^2,-q^3;q^5)_\infty^3(q,q^9;q^{10})_\infty$, then $c_{5n+3}=c_{5n+4}=0$ and $d_{5n+3}=d_{5n+4}=0$. In this paper, we prove that $a_{5n}=b_{5n+2}$, $a_{5n+1}=b_{5n+3}$, $a_{5n+2}=b_{5n+4}$, $a_{5n-1}=b_{5n+1}$, $c_{5n+3}=d_{5n+3}$, $c_{5n+4}=d_{5n+4}$, $c_{5n}=d_{5n}$, $c_{5n+2}=d_{5n+2}$, and $c_{5n+1}>d_{5n+1}$. We also record some other comparable results not listed by Tang.
Reference graph
Works this paper leans on
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[3]
Baruah, N.D., Begum, N.M.: Exact generating functions for the nu mber of partitions into distinct parts. Int. J. Number Theory 14, 1995–2011 (2018)
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Andrews, G.E., Bressoud, D.: Vanishing coefficients in infinite produ ct expansions. J. Aust. Math. Soc. Ser. A 27, 199–202 (1979)
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Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
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Hirschhorn, M.D.: Two remarkable q-series expansions. Ramanujan J. https://doi.org/ 10.1007/ s11139-018-0016-9 (2018)
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Mc Laughlin, J.: Further results on vanishing coefficients in infinite p roduct expansions. J. Aust. Math. Soc. Ser. A 98, 69–77 (2015)
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[8]
Tang, D.: Vanishing coefficients in some q-series expansions. Int. J. Number Theory https://doi.org/10.1142/S1793042119500398 (2018) Department of Mathematical Sciences, Tezpur University, N apaam-784028, Sonit- pur, Assam, INDIA E-mail address : nayan@tezu.ernet.in Department of Mathematical Sciences, Tezpur University, N apaam-784028, Sonit- pur, Assam, I...
Reviewed August 14, 2026 · model on record in the stance chip above.
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