Evaluation of two determinants involving $q$-integers

Zhi-Wei Sun

arxiv: 2605.16240 · v2 · pith:SW6ZZXOVnew · submitted 2026-05-15 · 🧮 math.CO · math.NT

Evaluation of two determinants involving q-integers

Zhi-Wei Sun This is my paper

Pith reviewed 2026-05-20 16:27 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords q-analoguesdeterminantsJacobi symbolfloor functionceiling functiondiscrete Fourier transformsodd integers

0 comments

The pith

Two determinants of q-analogue floor and ceiling matrices equal plus or minus a(a+1)/n times a power of q, multiplied by the Jacobi symbol.

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

The paper establishes closed-form evaluations for two n-by-n determinants whose entries are the q-analogues of floor((a j - (a+1)k)/n) and ceiling(((a+1)j - a k)/n). Using discrete Fourier transforms, it shows that the first determinant equals -(a(a+1)/n) q to the power (1-3n)/2 and the second equals +(a(a+1)/n) q to the power (n-1)/2, where n is a positive odd integer and (·/n) denotes the Jacobi symbol. A sympathetic reader would care because these give explicit formulas that replace direct determinant computation in settings where q-integers and floor or ceiling expressions arise. The identities hold for any integer a.

Core claim

The paper claims that for positive odd integer n and integer a, det[[ [floor((a j - (a+1) k)/n)]_q ]]_{1≤j,k≤n} equals -(a(a+1)/n) q^{(1-3n)/2} and det[[ [ceil(((a+1)j - a k)/n)]_q ]] equals (a(a+1)/n) q^{(n-1)/2}, with the Jacobi symbol appearing in both right-hand sides.

What carries the argument

Discrete Fourier transforms applied directly to the q-analogue matrix entries built from floor and ceiling expressions.

If this is right

The determinants admit simple closed forms that do not require expanding the n x n matrices.
The sign difference between the two identities is fixed by the choice of exponent on q and the overall sign.
The Jacobi symbol (a(a+1)/n) controls the value for different a, vanishing when a(a+1) is a quadratic non-residue modulo n.
Special cases for particular a yield determinants equal to zero or to explicit powers of q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The Fourier-transform technique might extend to other linear combinations inside the floor or ceiling arguments.
Similar determinant identities could exist for even n or for round instead of floor and ceiling.
These evaluations may simplify certain generating functions or partition identities that involve q-integers of this form.

Load-bearing premise

The discrete Fourier transform method applied to these particular q-floor and q-ceiling entries produces the stated closed forms when n is positive and odd.

What would settle it

For n=3 and a=1, expand both 3x3 determinants symbolically in q and check whether the resulting polynomials match the claimed right-hand sides numerically at q=2.

read the original abstract

The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\det\left[\left[\left\lfloor\frac{aj-(a+1)k}n\right\rfloor\right]_q\right]_{1\leqslant j,k\leqslant n}=-\left(\frac{a(a+1)}n\right)q^{(1-3n)/2}$$ and $$\det\left[\left[\left\lceil\frac{(a+1)j-ak}n\right\rceil\right]_q\right]_{1\leqslant j,k\leqslant n}=\left(\frac{a(a+1)}n\right)q^{(n-1)/2},$$ where $(\frac{\cdot}n)$ denotes the Jacobi symbol.

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

The paper gives explicit closed forms for two specific q-determinant identities via DFT, but the statements as written fail for n=1.

read the letter

The main takeaway is that this short note derives closed forms for the determinants of two n by n matrices whose entries are q-analogues of floor((a j - (a+1) k)/n) and ceiling((a+1) j - a k)/n, with the results involving a(a+1)/n times a power of q and a Jacobi symbol. The derivations rely on discrete Fourier transforms applied to the matrix entries, and the identities are presented as holding for any integer a and any positive odd integer n. That is the concrete new content: two previously unevaluated determinant formulas in the q-setting. The approach is straightforward for this corner of q-series and combinatorial number theory, and the explicit right-hand sides are the sort of thing that can be checked numerically for small odd n greater than 1. The paper earns credit for producing reproducible-looking identities rather than just asymptotic statements. The soft spot is the range of validity. The abstract states the claims for positive odd n without further restrictions, yet n=1 is included and the left-hand sides become independent of a while the right-hand sides scale with a(a+1). Direct substitution shows mismatch for a=1 and n=1 on both identities. Either the formulas need an extra condition such as n>1 or the derivation contains an implicit assumption not stated in the abstract. The Jacobi symbol and the q-powers look formally consistent once that is fixed, but the current write-up leaves the claim overstated. This is the kind of paper that specialists in q-analogues and determinant evaluations might want to see, especially if they are looking for concrete identities to test or extend. It is not broad enough to reorganize the field, but the explicit results could be cited in follow-up work on similar matrices. I would bring it to a reading group for the DFT technique and the numerical checks. It deserves peer review once the authors clarify the conditions on n and a; the core method appears sound enough to warrant referee time rather than a desk rejection.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to evaluate two n x n determinants whose entries are q-analogues of floor((a j - (a+1) k)/n) and ceil(((a+1) j - a k)/n) for integer a and positive odd integer n. Using discrete Fourier transforms on the matrix entries, the authors derive closed forms - (a(a+1)/n) q^{(1-3n)/2} and (a(a+1)/n) q^{(n-1)/2}, respectively, where (·/n) denotes the Jacobi symbol.

Significance. If the derivations are correct, the results would supply explicit evaluations of these particular q-determinants, extending known techniques for circulant or Toeplitz matrices to the q-setting. The DFT approach is a natural tool for such problems and, when successful, yields parameter-free closed forms that are falsifiable by direct computation for small odd n.

major comments (2)

[Abstract] Abstract (and the stated identities): the claim is made for every positive odd integer n, including n=1. For the first identity with n=1 the left-hand side is the 1x1 matrix whose entry is floor((a·1-(a+1)·1)/1) = -1, so [-1]_q = -q^{-1}, independent of a. The right-hand side is -(a(a+1)/1) q^{(1-3)/2} = -a(a+1) q^{-1}. These agree only when a(a+1)=1, contradicting the general claim (e.g., a=2 yields LHS=-q^{-1} vs. RHS=-6 q^{-1}).
[Abstract] Abstract (second identity): for n=1 the left-hand side is [ceil((a+1)·1 - a·1)/1]_q = [1]_q = 1, while the right-hand side is (a(a+1)/1) q^{(1-1)/2} = a(a+1). These are equal only for a=0 or a=-1, again contradicting the stated generality.

minor comments (1)

The manuscript should explicitly state whether a is restricted (e.g., positive or coprime to n) or whether the identities are intended only for n>1; the current wording leaves the domain ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the inconsistencies in the n=1 cases for both identities. We fully agree with these observations and will revise the manuscript to correct the stated range of n.

read point-by-point responses

Referee: [Abstract] Abstract (and the stated identities): the claim is made for every positive odd integer n, including n=1. For the first identity with n=1 the left-hand side is the 1x1 matrix whose entry is floor((a·1-(a+1)·1)/1) = -1, so [-1]_q = -q^{-1}, independent of a. The right-hand side is -(a(a+1)/1) q^{(1-3)/2} = -a(a+1) q^{-1}. These agree only when a(a+1)=1, contradicting the general claim (e.g., a=2 yields LHS=-q^{-1} vs. RHS=-6 q^{-1}).

Authors: The referee is correct. For n=1, the determinant reduces to a single entry [-1]_q = -q^{-1}, whereas the proposed closed form involves the factor a(a+1), leading to a mismatch for general a. We will modify the abstract and the theorem to state that n is a positive odd integer with n > 1. This restriction aligns with the assumptions likely implicit in the discrete Fourier transform approach used in the proofs. revision: yes
Referee: [Abstract] Abstract (second identity): for n=1 the left-hand side is [ceil((a+1)·1 - a·1)/1]_q = [1]_q = 1, while the right-hand side is (a(a+1)/1) q^{(1-1)/2} = a(a+1). These are equal only for a=0 or a=-1, again contradicting the stated generality.

Authors: We acknowledge this point as well. The second identity similarly fails for n=1, as the left-hand side is [1]_q = 1 while the right-hand side equals a(a+1). We will update the manuscript to exclude n=1, ensuring the claims are accurate for the intended cases. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external DFT to matrix entries

full rationale

The paper states that the two determinant identities are established via discrete Fourier transforms applied directly to the given q-analogue matrix entries defined from floor and ceil expressions. No quoted step reduces the claimed closed forms to a fitted parameter, self-definition, or load-bearing self-citation chain; the DFT step is presented as an independent computational tool whose output is the determinant value. The derivation therefore remains self-contained against external mathematical benchmarks and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The identities rest on standard properties of q-integers, the Jacobi symbol, and the applicability of discrete Fourier transforms to these matrix constructions; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard algebraic properties of the q-integer [m]_q = (1 - q^m)/(1 - q) and the Jacobi symbol.
Invoked directly in the statement of both determinant identities.
domain assumption Discrete Fourier transforms can be applied to evaluate the given q-analogue matrices.
Cited as the method used to establish the identities.

pith-pipeline@v0.9.0 · 5675 in / 1321 out tokens · 43447 ms · 2026-05-20T16:27:42.185738+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Via discrete Fourier transforms, we establish the following two identities: det[[ [floor( (a j - (a+1) k)/n ) ]_q ]] = − (a(a+1)/n) q^{(1−3n)/2}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Let n be a positive odd integer... det[[ [ceil( ((a+1) j - a k)/n ) ]_q ]] = (a(a+1)/n) q^{(n−1)/2}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Brunyate and P

A. Brunyate and P. L. Clark,Extending the Zolotarev-Frobenius approach to quadratic reciprocity, Ramanujan J.37(2015), 25–50

work page 2015
[2]

S. Fu, Z. Lin and Z.-W. Sun,Permanent identities, combinatorial sequences and permutation statistics, Adv. Appl. Math.163(2025), Article 102789

work page 2025
[3]

Huang and H

C. Huang and H. Pan,A remark on Zolotarev’s theorem, Colloq. Math.171(2023), 159–166

work page 2023
[4]

Ireland and M

K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, 2nd Edition, Grad. Texts. Math., vol. 84, Springer, New York, 1990

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Paltorp,The matrix determinant lemma, 2024

M. Paltorp,The matrix determinant lemma, 2024. Available from the website https://mipals.github.io/pubs/matrix/matrix determinant lemma

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Sun,On some determinants with Legendre symbol entries, Finite Fields Appl

Z.-W. Sun,On some determinants with Legendre symbol entries, Finite Fields Appl. 56(2019), 285–307

work page 2019
[7]

Sun,Evaluatedet[[⌊ aj−(a+1)k n ⌋]q]1⩽j,k⩽n anddet[[⌈ (a+1)j−ak n ⌉]q]1⩽j,k⩽n, Ques- tion 404733 at MathOverflow, Sept

Z.-W. Sun,Evaluatedet[[⌊ aj−(a+1)k n ⌋]q]1⩽j,k⩽n anddet[[⌈ (a+1)j−ak n ⌉]q]1⩽j,k⩽n, Ques- tion 404733 at MathOverflow, Sept. 24, 2021. Available from the website https://mathoverflow.net/questions/404733

work page 2021
[8]

Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021

Z.-W. Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021

work page arXiv 2021
[9]

Sun,On some determinants involving the tangent function, Ramanujan J.64 (2024), 309–332

Z.-W. Sun,On some determinants involving the tangent function, Ramanujan J.64 (2024), 309–332

work page 2024
[10]

Vrabel, A note on the matrix determinant lemma, Int

R. Vrabel, A note on the matrix determinant lemma, Int. J. Pure Appl. Math.111 (2016), 643–646

work page 2016
[11]

Wang and H.-L

L.-Y. Wang and H.-L. Wu,On the cyclotomic fieldQ(e 2πi/p and Zhi-Wei Sun’s con- jecture ondetM p, Finite Fields Appl.101(2025), Article 102533

work page 2025
[12]

Zolotarev,Nouvelle d´ emonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann

G. Zolotarev,Nouvelle d´ emonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann. Math.11(1872), 354–362. School of Mathematics, Nanjing University, Nanjing 210093, People’s Re- public of China Email address:zwsun@nju.edu.cn

work page

[1] [1]

Brunyate and P

A. Brunyate and P. L. Clark,Extending the Zolotarev-Frobenius approach to quadratic reciprocity, Ramanujan J.37(2015), 25–50

work page 2015

[2] [2]

S. Fu, Z. Lin and Z.-W. Sun,Permanent identities, combinatorial sequences and permutation statistics, Adv. Appl. Math.163(2025), Article 102789

work page 2025

[3] [3]

Huang and H

C. Huang and H. Pan,A remark on Zolotarev’s theorem, Colloq. Math.171(2023), 159–166

work page 2023

[4] [4]

Ireland and M

K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, 2nd Edition, Grad. Texts. Math., vol. 84, Springer, New York, 1990

work page 1990

[5] [5]

Paltorp,The matrix determinant lemma, 2024

M. Paltorp,The matrix determinant lemma, 2024. Available from the website https://mipals.github.io/pubs/matrix/matrix determinant lemma

work page 2024

[6] [6]

Sun,On some determinants with Legendre symbol entries, Finite Fields Appl

Z.-W. Sun,On some determinants with Legendre symbol entries, Finite Fields Appl. 56(2019), 285–307

work page 2019

[7] [7]

Sun,Evaluatedet[[⌊ aj−(a+1)k n ⌋]q]1⩽j,k⩽n anddet[[⌈ (a+1)j−ak n ⌉]q]1⩽j,k⩽n, Ques- tion 404733 at MathOverflow, Sept

Z.-W. Sun,Evaluatedet[[⌊ aj−(a+1)k n ⌋]q]1⩽j,k⩽n anddet[[⌈ (a+1)j−ak n ⌉]q]1⩽j,k⩽n, Ques- tion 404733 at MathOverflow, Sept. 24, 2021. Available from the website https://mathoverflow.net/questions/404733

work page 2021

[8] [8]

Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021

Z.-W. Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021

work page arXiv 2021

[9] [9]

Sun,On some determinants involving the tangent function, Ramanujan J.64 (2024), 309–332

Z.-W. Sun,On some determinants involving the tangent function, Ramanujan J.64 (2024), 309–332

work page 2024

[10] [10]

Vrabel, A note on the matrix determinant lemma, Int

R. Vrabel, A note on the matrix determinant lemma, Int. J. Pure Appl. Math.111 (2016), 643–646

work page 2016

[11] [11]

Wang and H.-L

L.-Y. Wang and H.-L. Wu,On the cyclotomic fieldQ(e 2πi/p and Zhi-Wei Sun’s con- jecture ondetM p, Finite Fields Appl.101(2025), Article 102533

work page 2025

[12] [12]

Zolotarev,Nouvelle d´ emonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann

G. Zolotarev,Nouvelle d´ emonstration de la loi de r´ eciprocit´ e de Legendre, Nouvelles Ann. Math.11(1872), 354–362. School of Mathematics, Nanjing University, Nanjing 210093, People’s Re- public of China Email address:zwsun@nju.edu.cn

work page