Further Applications of Cubic $q$-Binomial Transformations

Alexander Berkovich; Aritram Dhar

arxiv: 2510.00130 · v3 · submitted 2025-09-30 · 🧮 math.NT · math.CO

Further Applications of Cubic q-Binomial Transformations

Alexander Berkovich , Aritram Dhar This is my paper

Pith reviewed 2026-05-18 10:58 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords q-binomial coefficientsnon-negative coefficientscubic transformationsRogers-Szegő polynomialsq-series identitiespositivitypartition generating functions

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The pith

For parameters tied to powers of three, certain alternating q-binomial sums expand to polynomials with non-negative coefficients.

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

The paper proves that two families of the generating function G(N,M;α,β,K,q), when the parameters are chosen using a non-negative integer t and fractions with denominator three, are polynomials in q whose coefficients are all non-negative. A reader would care because this positivity removes the alternating signs in the original sum and often reveals a direct counting interpretation. The authors reach this conclusion by feeding the chosen parameters into known cubic positivity-preserving transformations and Rogers-Szegő polynomial identities. They also obtain explicit new identities that equate a finite positive sum to an infinite alternating sum of q-binomials.

Core claim

For non-negative integers n and t the sums G(n,n;4/3 + 3(3^t-1)/2, 5/3 + 3(3^t-1)/2, 3^{t+1}, q) and G(n-(3^t-1)/2, n+(3^t+1)/2; 8/3 + 2(3^t-1), 4/3 -(3^t-1), 3^{t+1}, q) are polynomials in q with non-negative coefficients. The proof proceeds by applying the cubic positivity-preserving transformations of Berkovich and Warnaar together with known Rogers-Szegő formulae to these specific arithmetic-progression choices of the parameters α, β and K.

What carries the argument

Cubic positivity-preserving transformations that convert an alternating sum of q-binomials into an equivalent sum whose terms are visibly non-negative when the parameters lie in certain arithmetic progressions built from powers of three.

Load-bearing premise

The cubic positivity-preserving transformations and the Rogers-Szegő formulae apply directly to the fractional and exponential parameter values chosen in the paper.

What would settle it

Compute the power series expansion in q of the first displayed G sum for n=1 and t=0; the appearance of any negative coefficient would refute the claim.

read the original abstract

Consider \begin{align*} G(N,M;\alpha,\beta,K,q) = \sum\limits_{j\in\mathbb{Z}}(-1)^jq^{\frac{1}{2}Kj((\alpha+\beta)j+\alpha-\beta)}\left[\begin{matrix}M+N\\N-Kj\end{matrix}\right]_{q}. \end{align*} In this paper, we prove the non-negativity of coefficients of some cases of $G(N,M;\alpha,\beta,K,q)$. For instance, for non-negative integers $n$ and $t$, we prove that\\ \begin{align*} G\left(n,n;\frac{4}{3}+\frac{3(3^t-1)}{2},\frac{5}{3}+\frac{3(3^t-1)}{2},3^{t+1},q\right) \end{align*} and \begin{align*} G\left(n-\frac{3^t-1}{2},n+\frac{3^t+1}{2};\frac{8}{3}+2(3^t-1),\frac{4}{3}-(3^t-1),3^{t+1},q\right)\\ \end{align*} are polynomials in $q$ with non-negative coefficients. Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szeg\"{o} polynomials, we establish new identities such as\\ \begin{align*} \sum\limits_{0\le 3j\le n}\dfrac{(q^3;q^3)_{n-j-1}(1-q^{2n})q^{3j^2}}{(q;q)_{n-3j}(q^6;q^6)_{j}} = \sum\limits_{j=-\infty}^{\infty}(-1)^jq^{6j^2}{2n\brack n-3j}_q. \end{align*}

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

Berkovich and Dhar apply cubic transformations to two new 3^t parameter families and claim non-negativity for the resulting G polynomials, though the negative beta case in the second family is the part worth double-checking.

read the letter

The main point is that this paper takes the cubic positivity-preserving transformations from Berkovich and Warnaar and plugs in specific parameter choices built around powers of 3 to get two fresh families where G(N,M; alpha, beta, K, q) has non-negative coefficients for non-negative integers n and t. They also pull out a few explicit summation identities as a side result, like the one relating a sum with (q^3;q^3) factors to an alternating q-binomial sum. That is the actual new content: concrete instances that were not listed in the earlier literature they cite. The first family keeps parameters positive, so that direction follows the known machinery without much extra work. The identities themselves are clean and could be handy for people who need explicit examples in this corner of q-series. The soft spot is the second family, where beta equals 4/3 minus (3^t minus 1) and turns negative right away for t equals 1 and worse after that. The original transformations and Rogers-Szegő base cases are usually stated for non-negative or positive parameters to keep the coefficient signs under control. If the paper simply substitutes without additional verification or cancellation arguments for the negative regime, the non-negativity claim rests on an untested extension. That is the one place the argument could be tightened. This is narrow-scope work aimed at specialists already tracking q-analogues and basic hypergeometric positivity results. Someone who knows the Berkovich-Warnaar papers will see the value immediately and can check the substitutions themselves. It is honest and incremental rather than broad, but the claims are specific enough to deserve a serious referee who can look at the proof details and the negative-beta handling. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that for non-negative integers n and t, the two families G(n,n; 4/3 + 3(3^t-1)/2, 5/3 + 3(3^t-1)/2, 3^{t+1}, q) and G(n-(3^t-1)/2, n+(3^t+1)/2; 8/3 + 2(3^t-1), 4/3 - (3^t-1), 3^{t+1}, q) are polynomials in q with non-negative coefficients. The proofs rely on cubic positivity-preserving transformations of Berkovich and Warnaar together with known Rogers-Szegő formulae; as applications, new identities are derived such as the displayed equality between a finite sum involving (q^3;q^3) and (q^6;q^6) Pochhammers and an alternating sum of q-binomials.

Significance. If the non-negativity claims hold under the stated parameter choices, the work supplies concrete new families of positive q-polynomials obtained via cubic transformations, extending earlier applications in the literature. The explicit new identity provides a falsifiable prediction that can be checked computationally for small n.

major comments (2)

[Abstract] Abstract (second family): β = 4/3 − (3^t − 1) equals −2/3 for t=1 and becomes more negative for t>1. The Berkovich–Warnaar cubic transformations and the Rogers–Szegő base cases are typically stated under hypotheses that include non-negative (or positive) real parameters to guarantee non-negative coefficients in the output. The manuscript must explicitly confirm that these transformations continue to preserve non-negativity when β<0, or provide a separate argument that any negative coefficients introduced are cancelled by the alternating sum in the definition of G.
[Abstract] Abstract (second family): for the non-negativity statement to be meaningful, N = n − (3^t − 1)/2 must be a non-negative integer. While this holds for sufficiently large n, the paper should state the precise lower bound on n in terms of t (or verify that the transformation hypotheses remain valid when N is allowed to be zero or small).

minor comments (2)

[Abstract] The second displayed G expression is split across two lines in a way that obscures the parameter list; reformat for readability.
[The displayed identity] In the new identity, confirm that the summation condition 0 ≤ 3j ≤ n together with the Pochhammer symbols (q^3;q^3)_{n-j-1} and (q^6;q^6)_j are defined for all integers j in the range (including when n-j-1 < 0).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and agree that the manuscript requires revisions for clarity on parameter ranges and transformation applicability.

read point-by-point responses

Referee: [Abstract] Abstract (second family): β = 4/3 − (3^t − 1) equals −2/3 for t=1 and becomes more negative for t>1. The Berkovich–Warnaar cubic transformations and the Rogers–Szegő base cases are typically stated under hypotheses that include non-negative (or positive) real parameters to guarantee non-negative coefficients in the output. The manuscript must explicitly confirm that these transformations continue to preserve non-negativity when β<0, or provide a separate argument that any negative coefficients introduced are cancelled by the alternating sum in the definition of G.

Authors: We agree that β is negative for t ≥ 1. The proofs apply the Berkovich–Warnaar transformations to Rogers–Szegő polynomials with the specific parameter choices in the manuscript. Although standard statements of the transformations assume non-negative parameters, the resulting G expression remains non-negative because the alternating sum in its definition cancels any potential negative contributions arising from β < 0. We will revise the manuscript to add an explicit paragraph confirming this cancellation mechanism, supported by the structure of the cubic transformation and verification for small t. revision: yes
Referee: [Abstract] Abstract (second family): for the non-negativity statement to be meaningful, N = n − (3^t − 1)/2 must be a non-negative integer. While this holds for sufficiently large n, the paper should state the precise lower bound on n in terms of t (or verify that the transformation hypotheses remain valid when N is allowed to be zero or small).

Authors: We thank the referee for this observation. For integer t ≥ 0, (3^t − 1)/2 is always an integer, so N is an integer precisely when n is an integer. To ensure N ≥ 0 we require n ≥ (3^t − 1)/2. The manuscript states the result for non-negative integers n and t but implicitly assumes n is large enough; we will revise the abstract and theorem statements to include the explicit lower bound n ≥ (3^t − 1)/2. When N = 0 the sum G reduces to a single non-negative term and the transformation hypotheses remain valid. revision: yes

Circularity Check

1 steps flagged

Self-citation to Berkovich-Warnaar cubic transformations is load-bearing for non-negativity but prior proofs remain independent

specific steps

self citation load bearing [Abstract]
"Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szegő polynomials, we establish new identities such as [identity] ... we prove that G(n,n;4/3 + 3(3^t-1)/2, ...) and G(n-(3^t-1)/2, ...) are polynomials in q with non-negative coefficients."

The non-negativity claim for the indicated G instances is justified by direct appeal to the Berkovich-Warnaar transformations whose authors overlap with the present paper; while the prior proofs are external, the load-bearing step for the new parameter choices reduces to this self-cited machinery rather than an independent derivation performed here.

full rationale

The paper's central non-negativity results for the two families of G(N,M;α,β,K,q) are obtained by applying the cubic positivity-preserving transformations from Berkovich-Warnaar (2019 or earlier) together with Rogers-Szegő identities. This constitutes a self-citation because Berkovich is a co-author of the present work. However, the transformations themselves are established in prior literature with their own proofs, and the current paper performs new applications to specific fractional/exponential parameters without re-deriving or fitting the transformations inside this manuscript. No self-definitional loop, fitted-input prediction, or ansatz smuggling occurs; the derivation chain therefore retains independent mathematical content outside the present text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of previously published cubic positivity-preserving transformations and on standard summation formulae for Rogers-Szegő polynomials; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Cubic positivity preserving transformations of Berkovich and Warnaar apply to the chosen parameter sets
Invoked to convert the alternating sum into a non-negative polynomial
domain assumption Known Rogers-Szegő polynomial formulae hold for the relevant q-Pochhammer products
Used to establish the new identities

pith-pipeline@v0.9.0 · 5879 in / 1311 out tokens · 50572 ms · 2026-05-18T10:58:20.752231+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Berkovich and A

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Berkovich, S

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M. E. H. Ismail, D. Kim, D. Stanton, Lattice paths and positive trigonometric sums,Constr. Approx.15(1999) 69–81

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Wang, Analytic proof of the Borwein conjecture,Adv

C. Wang, Analytic proof of the Borwein conjecture,Adv. Math.394(2022) 108028 54 pp

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Wang and C

C. Wang and C. Krattenthaler, An asymptotic approach to Borwein-type sign pattern theorems, preprint, arXiv:2201.12415

work page arXiv
[14]

S. O. Warnaar, The generalized Borwein conjecture. I. The Burge transform, in: B. C. Berndt, K. Ono (Eds.), q-Series with Applications to Combinatorics, Number Theory and Physics, in: Contemp. Math, vol. 291, AMS, Providence, RI, 2001, pp. 243–267

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S. O. Warnaar, The generalized Borwein conjecture. II. Refinedq-trinomial coefficients,Discrete Math.272 (2-3) (2003) 215–258. DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611, USA Email address:alexb@ufl.edu DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611, USA Email address:aritramdhar@ufl.edu

work page 2003

[1] [1]

G. E. Andrews, Partitions and the Gaussian sum, The mathematical heritage of C. F. Gauss, World Sci. Publ., River Edge, NJ, 1991, pp.35–42

work page 1991

[2] [2]

G. E. Andrews, On a Conjecture of Peter Borwein,J. Symbolic Comput.20 (5-6) (1995) 487–501

work page 1995

[3] [3]

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998

work page 1998

[4] [4]

G. E. Andrews, R. J. Baxter, D. M. Bressoud, W. H. Burge, P. J. Forrester, G. Viennot, Partitions with prescribed hook differences,Eur. J. Comb.8 (4) (1987) 341–350

work page 1987

[5] [5]

Berkovich, Some new positive observations,Discrete Math.343 (11) (2020) 112040, 8 pp

A. Berkovich, Some new positive observations,Discrete Math.343 (11) (2020) 112040, 8 pp

work page 2020

[6] [6]

Berkovich, Bressoud’s identities for even moduli

A. Berkovich, Bressoud’s identities for even moduli. New companions and related positivity results,Discrete Math.345 (12) (2022) 113104, 13 pp

work page 2022

[7] [7]

Berkovich and A

A. Berkovich and A. Dhar, Extension of Bressoud’s generalization of Borwein’s Conjecture and some exact results,Ramanujan J.67 (16), 2025 (to appear in the special issue in honor of the 85th birthdays of George E. Andrews and Bruce C. Berndt)

work page 2025

[8] [8]

Berkovich, S

A. Berkovich, S. O. Warnaar, Positivity preserving transformations forq-binomial coefficients,Trans. Am. Math. Soc.357 (6)(2005) 2291–2351

work page 2005

[9] [9]

D. M. Bressoud, Some identities for terminatingq-series,Math. Proc. Camb. Philos. Soc.89(1981) 211–223

work page 1981

[10] [10]

D. M. Bressoud, The Borwein conjecture and partitions with prescribed hook differences,Electron. J. Combin.3 (2)(1996) #4. 11

work page 1996

[11] [11]

M. E. H. Ismail, D. Kim, D. Stanton, Lattice paths and positive trigonometric sums,Constr. Approx.15(1999) 69–81

work page 1999

[12] [12]

Wang, Analytic proof of the Borwein conjecture,Adv

C. Wang, Analytic proof of the Borwein conjecture,Adv. Math.394(2022) 108028 54 pp

work page 2022

[13] [13]

Wang and C

C. Wang and C. Krattenthaler, An asymptotic approach to Borwein-type sign pattern theorems, preprint, arXiv:2201.12415

work page arXiv

[14] [14]

S. O. Warnaar, The generalized Borwein conjecture. I. The Burge transform, in: B. C. Berndt, K. Ono (Eds.), q-Series with Applications to Combinatorics, Number Theory and Physics, in: Contemp. Math, vol. 291, AMS, Providence, RI, 2001, pp. 243–267

work page 2001

[15] [15]

S. O. Warnaar, The generalized Borwein conjecture. II. Refinedq-trinomial coefficients,Discrete Math.272 (2-3) (2003) 215–258. DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611, USA Email address:alexb@ufl.edu DEPARTMENT OFMATHEMATICS, UNIVERSITY OFFLORIDA, GAINESVILLE, FL 32611, USA Email address:aritramdhar@ufl.edu

work page 2003