A q-Supercongruence Motivated by Higher-Order Generalized Lehmer-Euler Numbers

Wei-Wei Qi

REVIEW 3 major objections 4 minor 18 references

A q-Supercongruence Motivated by Higher-Order Generalized Lehmer-Euler Numbers

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves a two-parameter family of q-supercongruences: for positive odd $n$ and positive even $\alpha \leq n$, a q-binomial sum modeled on higher-order generalized Lehmer-Euler numbers is congruent modulo $\Phi_n(q)^2$ to an…

desk verdict New parametric q-supercongruence, but the proof hinges on auxiliary congruences that are false for composite odd n. read the letter →

arxiv 2507.16825 v1 pith:2JQIPPNM submitted 2025-07-05 math.CO

Wei-Wei Qi This is my paper · ORCID

classification math.CO MSC 11A0711B6833D15

keywords GeneralizedLehmer-Eulernumbersq-supercongruenceq-congruencecyclotomicpolynomialq-binomialcoefficientsEulerFermatquotientq-harmonicsums

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 a q-supercongruence for a polynomial built from the higher-order generalized Lehmer-Euler numbers. For every positive odd integer $n$ and positive even integer $\alpha \leq n$, the q-binomial sum on the left of (1.5) is congruent modulo $\Phi_n(q)^2$ to a closed expression made of q-integers, the q-Fermat quotient $Q_n(2,q)$, and alternating q-harmonic sums, and the companion congruence (1.6) gives the same evaluation for a nested version of the sum. The point of the congruence is that it is genuinely two-parametric and lives modulo the square of a cyclotomic polynomial: when $n$ is a prime $p$ and $q\to 1$, it reduces to a pair of congruences modulo $p^2$ that involve Euler numbers $E_{p-2}$ and the Fermat quotient $q_p(2)$. A sympathetic reader would care because it transfers the classical Lehmer-Euler number theory into the q-supercongruence setting, where similar results for other q-binomial supercongruences have been a central theme.

What carries the argument

The motivating object is the polynomial $M_n^*(\alpha)=\sum_{k=0}^{n}\binom{\alpha+k-1}{k}\binom{\alpha+n}{n-k}$, whose q-analogue is the sum evaluated in Theorem 1.1. The proof machinery has three parts: the q-binomial congruence (2.4), which rewrites the product $\binom{\alpha+k-1}{k}_q\binom{\alpha+n-1}{n-1-k}_q$ modulo $\Phi_n(q)^2$ as a multiple of $(1-q^n)/(1-q^{k+\alpha})$; the boundary product identities (2.6)-(2.8), which evaluate the corner terms of the sum; and the q-harmonic congruences (2.1)-(2.2), stated without proof as weaker forms of results in [16], which replace alternating sums such as $\sum_{k=1}^{n-1}(-1)^k/(1-q^k)$ by combinations of $Q_n(2,q)$ and simple q-rational functions. The q-Fermat quotient $Q_n(2,q)=((q^2,q^2)_{n-1}/(q,q)_{n-1}-1)/[n]$ is the invariant that carries the modular information.

What would settle it

Evaluate the two sides of (1.5) for $n=5$, $\alpha=2$ as polynomials in $q$, reduce modulo $\Phi_5(q)^2=(q^4+q^3+q^2+q+1)^2$, and check equality; also verify (2.1) and (2.2) directly for $n=5$ and $n=9$. A mismatch in either check would settle that the statement, or the proof's stated q-harmonic input, is wrong.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for positive odd $n$ and positive even $\alpha$ with $\alpha \leq n$, the q-binomial sum in (1.5) is congruent modulo $\Phi_n(q)^2$ to an explicit closed form, and the companion congruence (1.6) gives the corresponding residue for the nested sum $\sum_{j=0}^{n-1}q^j\sum_{k=0}^{j}\cdots$. Displayed in the paper, (1.5) reads $$\sum_{k=0}^{n-1} $q^{{\binom{k+1}}${2}} \binom{\$\alpha$+k-1}{k}_q \binom{\$\alpha$+n-1}{n-1-k}_q \equiv \frac{2[n](1-q)+2q^\$\alpha$[n]-[\$\alpha$]}{[\$\alpha$]} - 2[n]Q_n(2,q) - 2[n]\sum_{k=1}^{\$\alpha$}\frac{(-1)^k}{[k]} \pmod{\Phi_n(q)^2}.$$ The proof rewrites q-binomial coefficients modulo $\Phi_n(q)^2$, uses the two q-harmonic congruences (2.1)-(2.2) to simplify alternating sums, and then collects terms; Corollary 1.2 specializes $n=p$ and $q\to 1$ to get congruences modulo $p^2$ involving Euler numbers $E_{p-2}$ and the Fermat quotient $q_p(2)$.

Load-bearing premise

The whole derivation depends on two auxiliary formulas about alternating sums of reciprocals of q-integers, formulas that are stated without proof and only loosely attributed to an earlier paper; if either formula is wrong or misquoted, the main congruence is not established.

Editorial extensions

If this is right

For prime $n=p$ and $q\to 1$, Theorem 1.1 gives the two congruences of Corollary 1.2 modulo $p^2$, written in terms of the Fermat quotient $q_p(2)$ and the Euler numbers $E_{p-2}$.
The second congruence (1.6) is a weighted-sum consequence of the first: summing the $k$-th term with weight $[n]-[k]$ gives exactly the nested expression, so the two formulas are two views of one q-evaluation.
The result supplies a two-parameter family of q-supercongruences modulo the square of a cyclotomic polynomial, covering all odd $n$ rather than only primes.
At $q=1$ and for $n=p$, the right-hand sides of the congruences become rational expressions that, through the Euler-number identity (1.7), are $p$-adic residues; Corollary 1.2 records these as binomial-sum congruences modulo $p^2$.

Reading between the lines

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

A likely extension, not claimed in the paper, is that the parity conditions are technical: the same reduction should work for even $n$ or odd $\alpha$ after adjusting the signs in the alternating sums.
If a full proof of the auxiliary congruences (2.1)-(2.2) for all odd $n$ is supplied, the same method should apply to other q-binomial sums that reduce to alternating q-harmonic series.
One concrete testable strengthening would be to simplify the right-hand side of (1.5) into a single finite q-hypergeometric evaluation, converting the congruence into a closed-form identity modulo $\Phi_n(q)^2$.

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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

Let me get to the point: the paper has a real new result, but the proof as written is not valid. The main congruence (1.5) and the companion (1.6) give a new two-parameter family of q-supercongruences, and the q→1 limit ties back to Lehmer-Euler numbers in a way that is cute rather than deep. The algebraic core—the manipulation from (2.4) to (2.11)—is standard and coherent, so the author knows the technique.

The trouble is in the two auxiliary congruences (2.1) and (2.2). They are stated for every odd n and dismissed as 'weaker versions' of Pan's results. Pan's theorems are for prime p. The extension to composite n is not a weakening; it is false. For n=9, a primitive root calculation gives Q_9(2,q)=0 mod Φ_9(q), so (2.2) would force sum_{k=1}^4 1/(1-q^{2k}) ≡ 0 mod Φ_9(q). At q=e^{2πi/9}, that sum is about 2.000-0.978i, not zero. So (2.2) is simply wrong. The paper also writes 1/(1-q^n) and 1/[n] in congruences modulo Φ_n(q) even though those denominators are not coprime to the modulus; under the paper's own convention, those expressions are not defined.

This is a load-bearing flaw: (2.3), (2.11), and (2.20) all pass through these congruences. The theorem may still be true—spot checks for n=9 and α=2,4,6,8 satisfy (1.5)—so the fix is probably either to prove the auxiliary congruences with cleared denominators or to restrict the statement to prime n. But as written, the proof does not establish the result for all odd n.

I would not desk-reject this. The sum is new, the parameter family is genuine, and the paper fits a live literature. A careful referee could either fix the auxiliary steps or help the author restrict the main theorem to a provable range. But I also would not accept it in its current form. Anyone citing this needs to check the congruence numerically first.

Bring it to reading group if you want a case study in how prime-only congruence methods do not automatically extend to composite moduli.

Referee Report

3 major / 4 minor

Summary. The paper defines a q-polynomial motivated by higher-order generalized Lehmer-Euler numbers and proves two q-supercongruences (Theorem 1.1) modulo the square of the n-th cyclotomic polynomial for odd n and even alpha with alpha <= n. The right-hand sides are explicit in q-integers, the q-Fermat quotient, and finite q-harmonic sums. Taking n = p prime and q to 1 yields two supercongruences for generalized Lehmer-Euler numbers (Corollary 1.2). The proof proceeds by manipulating q-binomial coefficients and q-harmonic sums, reducing the main sums to expressions involving two auxiliary congruences, (2.1) and (2.2), asserted in Section 2.

Significance. If the main result is correct, it provides a new parametric family of q-supercongruences with a concrete, non-fitted right-hand side, and it connects a classical object (Lehmer-Euler numbers) to contemporary q-congruence theory. The proof is largely elementary and self-contained apart from two auxiliary congruences. The paper also gives a q = 1 prime corollary that recovers congruences for the generalized Euler numbers of higher order. These are worthwhile contributions to the q-supercongruence literature, provided the proof is made rigorous.

major comments (3)

[Section 2, (2.1)-(2.2)] The two auxiliary congruences are stated for every odd integer n but are not proved. The text says only that they are weaker versions of Pan's results [16, Lemma 2.4 and Theorem 1.1], which are stated for prime p modulo [p]^2. The reduction from prime p to arbitrary odd n is not supplied, and it is not a formal consequence of the cited statements. Since these congruences are the first step in the proof of Theorem 1.1, the author should either prove (2.1) and (2.2) directly or give a precise reference that covers composite n. In fact, both congruences are true for all odd n and can be proved by elementary partial-fraction identities at primitive n-th roots; a short proof should be included.
[Section 2, (2.3) and (2.17)] The congruences (2.3) and (2.17) contain the terms 1/(1-q^n) and q^n/(1-q^n), respectively. The denominator 1-q^n is divisible by Phi_n(q), so these expressions are not well-defined under the paper's own convention stated in Section 1, which requires the denominator to be coprime to the modulus. Although the singular terms later cancel after multiplication by (1-q^n) in (2.5) and (2.15), the manuscript never articulates this clearing of denominators. The proof should be reformulated, for instance by deriving congruences for (1-q^n) times the involved sums, or by explicitly working in the localized ring with denominators coprime to Phi_n(q) and clearing all non-invertible factors. This is a load-bearing rigor gap because the derivation of (2.11) and (2.20) depends on these intermediate congruences.
[Theorem 1.1, right-hand sides] The statements (1.5) and (1.6) use expressions such as [n] Q_n(2,q) and [n] sum_{k=1}^alpha (-1)^k/[k]. The q-Fermat quotient Q_n(2,q) is defined with denominator [n], which is not coprime to Phi_n(q), and [k] is not invertible modulo Phi_n(q) when k = n. Although the alpha <= n condition with n odd and alpha even implies alpha < n, so the harmonic sum avoids k = n, the notation is still formally problematic. The author should define [n] Q_n(2,q) directly as (q^2;q^2)_{n-1}/(q;q)_{n-1} - 1, a rational function whose denominator is coprime to Phi_n(q)^2, and similarly specify the meaning of the harmonic-sum expressions after clearing denominators. Without such clarification, the congruences are not well-posed under the paper's stated convention.

minor comments (4)

[Abstract] The phrase "determine its a q-supercongruence" is ungrammatical; it should be "determine a q-supercongruence for it."
[Section 2, after (2.2)] The claim that (2.1) and (2.2) are "weaker versions" of Pan's results is misleading: Pan's congruences are modulo [p]^2 for prime p, whereas (2.1) and (2.2) are identities modulo Phi_n(q) for all odd n and are of a different nature. A direct proof or a different attribution would be more accurate.
[References] There are several typographical errors in the references: [2] should be "Lehmer-Euler number" rather than "Lember-Euler number"; [18] should be "Refinements" rather than "Reffnements"; and [5] lists three authors but the text refers to it as a two-author work by Gu and Guo.
[Equation (1.3)] The modulus in (1.3), written as [p]^2 q^{n/p}, is not standard notation and is not subsequently defined; please clarify the intended meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed q-supercongruences have an explicit right-hand side and the proof reduces them to external q-harmonic congruences, not to the theorem itself.

full rationale

I traced the derivation chain from Theorem 1.1 through Section 2. The right-hand sides of (1.5) and (1.6) are explicit expressions in [n], [α], q^α, Q_n(2,q), and finite q-harmonic sums. No parameter is fitted to force the congruence, and the left-hand side is not defined as the right-hand side. The proof obtains (2.11) by substituting (2.3), (2.8), and (2.10) into (2.5); these are algebraic manipulations of q-binomial coefficients together with the external congruences (2.1) and (2.2), attributed to Pan [16]. The first main congruence (1.5) then follows from (2.12) and (2.13), and (1.6) is derived from (1.5) together with further manipulations in (2.14)-(2.20). None of these steps assumes the target congruence as an input, and none defines a quantity in terms of the quantity being predicted. The author's self-citation [14] appears only in the introductory survey of q-congruence literature and is not load-bearing. The only flagged weakness is that (2.1) and (2.2) are asserted without proof, with the text saying they 'can be easily proved by the same method as them,' and their extension from prime p to arbitrary odd n may be doubtful; however, that is a correctness or rigor concern about an external lemma, not an instance of circular reasoning. The derivation is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central congruence rests on cited prior lemmas and standard q-congruence conventions; no free parameters are fitted, and no new postulates such as fictional particles or forces are introduced. The polynomial M_n^*(α) is a notation for a finite sum rather than an invented entity.

assumptions (3)

domain assumption The q-harmonic congruences (2.1) and (2.2) hold for odd n.
Stated without proof in Section 2; said to follow by the method of Pan [16]. They are load-bearing for (2.3) and (2.16).
standard math The Euler polynomial identity (1.7) and Fermat's little theorem as used in (1.8).
Used to pass from Theorem 1.1 to Corollary 1.2; cited to [17] and [18].
domain assumption q-congruence arithmetic with rational functions behaves as in the paper's definition, including cancellation of factors like (1-q^n) after multiplication.
The proof uses terms like 1/(1-q^n) that are not invertible modulo Φ_n(q) but become well-defined after multiplying by (1-q^n); this is standard in the q-congruence literature but not fully formalized here.

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Cite this review

Pith. "Pith review of A q-Supercongruence Motivated by Higher-Order Generalized Lehmer-Euler Numbers." pith.science (2026). https://pith.science/paper/2JQIPPNM

@misc{pith2026250716825,
  author       = {Pith},
  title        = {Pith review of: A q-Supercongruence Motivated by Higher-Order Generalized Lehmer-Euler Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JQIPPNM}},
  note         = {Machine review of arXiv:2507.16825}
}

read the original abstract

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized Lehmer-Euler numbers and determine its a q-supercongruence.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler

Lehmer, D. H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler. Ann. Math. 2(36), 637-–649 (1935)

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[2]

and Liu G.-D.: Congruence properties of Lember-Euler number

Komatsu T. and Liu G.-D.: Congruence properties of Lember-Euler number. Aequat. Math. 99, 1337–1355 (2025)

work page 2025
[3]

Tamkang J

Luo, Q.-M.: An explicit formula for the Euler numbers of higher order. Tamkang J. Math. 36(4), 315–317 (2005)

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[4]

Guo, V.J.W., Zeng, J.: New congruences for sums involving Ap´ ery numbers or central Delannoy numbers. Int. J. Number Theory 36, 2003–2016 (2012)

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and Guo, V.J.W., Zeng J.: q-Analogues of two supercongruences of Z.-W

Gu C.-Y. and Guo, V.J.W., Zeng J.: q-Analogues of two supercongruences of Z.-W. Sun. Czech. Math. J. 70, 757–765 (2020)

work page 2020
[6]

http : //math.ecnu.edu.cn/ jwguo/maths/quad.pd f(2021)

Guo V.J.W., Schlosser M.J., Zudilin W.: New quadratic identities for basic hypergeometric series and q-congruences. http : //math.ecnu.edu.cn/ jwguo/maths/quad.pd f(2021)

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and Jiang X.-T.: On the divisibility of sums of even powers of q-binomial coeffi- cients

Liu J.-C. and Jiang X.-T.: On the divisibility of sums of even powers of q-binomial coeffi- cients. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116(2):76 (2022)

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Shi, L.-L., Pan, H.: A q-analogue of Wolstenholme’s harmonic series congruence. Am. Math. Mon. 114, 529-–531 (2007)

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Rocky Mt

Pan, H.: A generalization of Wolstenholme’s harmonic series congruence. Rocky Mt. J. Math. 38, 1263—1269 (2008)

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and Zhang Y.: Ageneralization of Morley’s congruence

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Liu, J.-C.: On a congruence involving q-Catalan numbers. C. R. Math. Acad. Sci. Paris 358, 211-–215 (2020)

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2025 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler

Lehmer, D. H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler. Ann. Math. 2(36), 637-–649 (1935)

work page 1935

[2] [2]

and Liu G.-D.: Congruence properties of Lember-Euler number

Komatsu T. and Liu G.-D.: Congruence properties of Lember-Euler number. Aequat. Math. 99, 1337–1355 (2025)

work page 2025

[3] [3]

Tamkang J

Luo, Q.-M.: An explicit formula for the Euler numbers of higher order. Tamkang J. Math. 36(4), 315–317 (2005)

work page 2005

[4] [4]

Guo, V.J.W., Zeng, J.: New congruences for sums involving Ap´ ery numbers or central Delannoy numbers. Int. J. Number Theory 36, 2003–2016 (2012)

work page 2012

[5] [5]

and Guo, V.J.W., Zeng J.: q-Analogues of two supercongruences of Z.-W

Gu C.-Y. and Guo, V.J.W., Zeng J.: q-Analogues of two supercongruences of Z.-W. Sun. Czech. Math. J. 70, 757–765 (2020)

work page 2020

[6] [6]

http : //math.ecnu.edu.cn/ jwguo/maths/quad.pd f(2021)

Guo V.J.W., Schlosser M.J., Zudilin W.: New quadratic identities for basic hypergeometric series and q-congruences. http : //math.ecnu.edu.cn/ jwguo/maths/quad.pd f(2021)

work page 2021

[7] [7]

and Jiang X.-T.: On the divisibility of sums of even powers of q-binomial coeffi- cients

Liu J.-C. and Jiang X.-T.: On the divisibility of sums of even powers of q-binomial coeffi- cients. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116(2):76 (2022)

work page 2022

[8] [8]

Shi, L.-L., Pan, H.: A q-analogue of Wolstenholme’s harmonic series congruence. Am. Math. Mon. 114, 529-–531 (2007)

work page 2007