A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind

Abdelhay Benmoussa

REVIEW 2 major objections 4 minor 3 references

A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes identity (7): $\sum_{k=0}^n 2^{n-k}s(n,k)B_k = \sum_{k=0}^n b(n,k)(-1)^k k!/(k+1)$, a first-kind analogue of the classical Stirling–Bernoulli relation.

desk verdict Correct but not new: Theorem 2 is sound, but identity (7) is an immediate corollary of the already-published identity (9), so the novelty claim is substantially overstated. read the letter →

arxiv 2505.22819 v5 pith:VNVXLRIT submitted 2025-05-28 math.GM

Abdelhay Benmoussa This is my paper

classification math.GM MSC 11B6805A1811B7333C10

keywords BernoullinumberStirlingofthefirstkindBesseloperationalidentityFaulhaber'sformulapolynomialcombinatorial

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

This paper tries to establish a closed identity that ties together Bernoulli numbers, Stirling numbers of the first kind, and Bessel numbers of the first kind. The identity reads $\sum_{k=0}^n 2^{n-k}s(n,k)B_k=\sum_{k=0}^n b(n,k)(-1)^k k!/(k+1)$ and is put forward as the first-kind analogue of the classical formula $B_n=\sum_{k=0}^n S(n,k)(-1)^k k!/(k+1)$. The proof obtains a polynomial identity from an operational identity for the operator $x^{-1}D$, then extracts the coefficient of $m^1$ after summing over $t=0,\ldots,m-1$. If valid, the result reveals a formal duality between Stirling numbers of the two kinds and yields the corollary identity $\sum_{k=i}^n 2^{n-k}s(n,k)S(k,i)=b(n,i)$.

What carries the argument

The load-bearing mechanism is the operational identity $(x^{-1}D)^n=\sum_{k=0}^n b(n,k)x^{k-2n}D^k$, cited from the literature, which expresses repeated applications of the operator $x^{-1}D$ in terms of ordinary derivatives and defines the Bessel numbers as its coefficients. Applying it to $x^t$ turns it into the polynomial identity (8), and Lemma 1 rewrites the same product using Stirling numbers of the first kind with powers of 2. Summing $t=0,\dots,m-1$ and comparing the coefficient of $m^1$ — Faulhaber's formula on the left, the telescoping identity $\Delta(t)_{k+1}=(k+1)(t)_k$ plus Stirling expansion on the right — forces identity (7).

What would settle it

Evaluate identity (7) at $n=3$: with $B_0=1$, $B_1=-1/2$, $B_2=1/6$, $B_3=0$, $s(3,1)=2$, $s(3,2)=-3$, $s(3,3)=1$, and $b(3,k)$ from (4), both sides must equal $-5$; a different value at this or any small $n$ would refute the claimed identity.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: for every integer $n\ge0$, $\sum_{k=0}^n 2^{n-k}s(n,k)B_k = \sum_{k=0}^n b(n,k)(-1)^k k!/(k+1)$, with $s(n,k)$ the signed Stirling numbers of the first kind and $b(n,k)$ the Bessel numbers of the first kind defined by formula (4). The classical identity for Stirling numbers of the second kind has the same right-hand factor $(-1)^k k!/(k+1)$, which is what makes the result an analogue. The proof derives the polynomial identity $\prod_{j=0}^{n-1}(t-2j)=\sum_{k=0}^n b(n,k)(t)_k=\sum_{k=0}^n 2^{n-k}s(n,k)t^k$, sums over a block of integers, and compares the coefficient of $m^1$, using Faulhaber's formula on one side and falling-factorial telescoping with a Stirling expansion on the other. The paper also draws a corollary, $\sum_{k=i}^n 2^{n-k}s(n,k)S(k,i)=b(n,i)$, connecting Stirling numbers of both kinds to Bessel numbers.

Load-bearing premise

The proof depends on the cited operational identity (5), which the paper does not prove; if that identity or the stated formula (4) for the Bessel numbers were wrong, identity (8) and the theorem would collapse.

Editorial extensions

If this is right

For every $n\ge0$, identity (7) provides a first-kind counterpart to the classical Stirling–Bernoulli formula.
The polynomial identity (8) is valid for all $n$, giving a bridge between a product of even-spaced linear factors, the falling-factorial expansion, and the ordinary-power expansion.
Corollary 3 states $\sum_{k=i}^n 2^{n-k}s(n,k)S(k,i)=b(n,i)$ for $0\le i\le n$, a finite identity mixing Stirling numbers of both kinds with Bessel numbers.
The right-hand summand $(-1)^k k!/(k+1)$ is the same factor that appears in the classical second-kind identity, so the two identities are formal mirror images.

Reading between the lines

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

Going beyond the paper, the coefficient-of-$m^1$ extraction should transfer to any differential-operator identity of the form $(\phi(x)D)^n=\sum_k a(n,k)\psi_k(x)D^k$, so analogues of (7) may exist for other operator families.
Going beyond the paper, a $q$-deformation of Lemma 1's product $\prod_{j=0}^{n-1}(t-2j)$ is a natural next step that could produce a $q$-analogue of (7), though the paper does not explore this direction.
Going beyond the paper, the shared factor $(-1)^k k!/(k+1)$ in both the second-kind and first-kind formulas hints at an underlying inversion between the two kinds of Stirling numbers; making that inversion explicit would be a separate project.

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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

Short version: the main proof is correct, but the central claim of a new identity does not survive contact with the literature. Identity (7) follows in two lines from the cited identity (9) in [3] plus the classical Bernoulli expansion (1). The paper presents Corollary 3 as if (9) were being derived from (7), but that argument is invalid: from an equality of weighted sums with fixed coefficients (-1)^i i!/(i+1), you cannot conclude the summands are equal, because those coefficients are not linearly independent. The real implication is the reverse: (9) implies (7). So calling (7) a new structural connection is unsupported by the paper's own citations.

What is good: Lemma 1 is fine, applying the operator identity (5) to x^t gives the polynomial identity (8), and the coefficient-of-m^1 comparison using Faulhaber's formula and falling-factorial telescoping is clean. The operator identity (5) is only cited, but it is equivalent to the explicit formula (4) and can be checked by induction, so I would not treat that as a real gap.

The soft spots are, in order of importance: (1) the novelty framing needs to be scaled way back; the author should either prove that (7) is not already in [2] or [3], or reframe the paper as a self-contained direct proof of a known identity; (2) the corollary proof is logically flawed and should be corrected or removed; (3) the acknowledgment mentions a MathOverflow discussion, which is fine but does not change the novelty question.

Who is this for? Readers who enjoy operational derivations of classical number identities. It is a correct small result, not a breakthrough. The direct derivation has some pedagogical value even if the identity itself is known.

Recommendation: I would not desk-reject the underlying mathematics, but I would not accept the paper as is. It deserves a serious referee only if the author is willing to make major revisions: honestly reframe what is new, fix the corollary, and check [2] and [3] more carefully. If the author cannot establish that (7) is new, then the publishable contribution is the direct proof, not the identity.

Referee Report

2 major / 4 minor

Summary. The paper claims to establish a new identity (Equation 7) relating Bernoulli numbers, Stirling numbers of the first kind, and Bessel numbers of the first kind. The proof proceeds by deriving a polynomial identity (Equation 8) via an operational identity (Equation 5) and comparing coefficients of m^1 after summing from t=0 to m-1. The paper also states Corollary 3, an identity (Equation 9) involving Stirling numbers of both kinds and Bessel numbers.

Significance. The identity (7) is correct, and the derivation in Section 3 is coherent, relying on standard generating-function and operator identities. The proof is concise and, as a derivation, is valid. However, the claimed novelty is not established: as the paper's own Corollary 3 shows, identity (9) (cited to [3]) together with the classical expansion (1) immediately yields (7). Thus the paper's main contribution reduces to an alternative proof of a known identity rather than a new structural connection. The proof of Corollary 3 contains a logical gap, although the statement of (9) is known to be true from [3].

major comments (2)

[Section 1 and Corollary 3] The identity (7) is not new. The proof of Corollary 3 substitutes the classical expression B_k = sum_i S(k,i)(-1)^i i!/(i+1) into the left-hand side of (7) and, using identity (9) (which is cited from [3]), obtains the right-hand side of (7). Therefore (7) is a direct formal consequence of (9) and (1). The authors must either demonstrate that (9) does not imply (7), or revise the title, abstract, and introduction to present (7) as a known identity with an alternative proof rather than as a new identity.
[Corollary 3, proof] The inference from the equality of two weighted sums to the termwise identity (9) is invalid. From sum_i c_i A_i = sum_i c_i B_i with scalar coefficients c_i = (-1)^i i!/(i+1), one cannot conclude A_i = B_i for each i, because the c_i are not linearly independent in the scalar field. The conclusion (9) may be true, but the proof as written does not establish it; the authors should either give a correct proof or cite [3] directly for this identity.

minor comments (4)

[Section 2, Equation (4)] The definition of b(n,k) does not cover b(0,0), which is needed for the n=0 case in (7). Please specify b(0,0)=1 and b(n,0)=0 for n>0 consistently.
[Section 3, proof of Theorem 2] In the line after 'Applying (5) to the monomial x^t', the intermediate step showing the cancellation of the factor x^{t-2n} is omitted; including it would improve readability.
[Corollary 3, proof] The phrase 'Since the factors ... are independent' is unclear; the argument needs clarification or removal, especially because the preceding inference is not valid.
[Introduction] The paper does not mention identity (9) from [3] in the introduction; adding a reference there would help contextualize the novelty claim and prevent the appearance of overclaiming.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof of Theorem 2 is a direct deduction from Lemma 1, the external operator identity (5), and standard summation identities; no step assumes the target identity.

full rationale

The derivation chain is self-contained. Lemma 1 rewrites the product $\prod_{j=0}^{n-1}(t-2j)$ using the falling factorial expansion (3). Theorem 2 then applies the cited operator identity (5) of Han and Seo [2] to the monomial $x^t$, giving the polynomial identity (8) after cancellation of $x^{t-2n}$; summing (8) over $t=0,\ldots,m-1$ and comparing the coefficient of $m^1$ via Faulhaber's formula on the left and the telescoping identity $\Delta(t)_{k+1}=(k+1)(t)_k$ on the right yields exactly (7). None of these steps assumes (7), and (5) is an external result, not a self-citation and not a disguised form of the target identity. There are no fitted parameters or predicted quantities. The Corollary's attempt to conclude (9) from equality of two weighted sums is logically questionable because the scalar factors are not linearly independent in a single equation, and the novelty of (7) may be weakened if (9) already appears in [3]; these are correctness and novelty concerns, not circularity. Therefore the paper is not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard definitions and on a cited operator identity; the only imported load-bearing fact is (5), which is standard but unproved here.

assumptions (4)

standard math Bernoulli numbers are defined by the generating function t/(e^t-1) = sum B_n t^n/n!, and Faulhaber's formula gives the coefficient of m^1 in sum_{t=0}^{m-1} t^k as B_k.
Used in Section 3 to convert the summed left side of (8) into Bernoulli numbers.
standard math Stirling numbers of the first kind satisfy (x)_n = sum s(n,k) x^k, and (m)_{k+1} expands with s(k+1,1)=(-1)^k k!.
Used in Lemma 1 and in extracting the coefficient of m^1 from the right side.
domain assumption Bessel numbers satisfy the operational identity (5) cited from [2].
This identity is the bridge that produces the polynomial identity (8). The paper does not prove it.
standard math The classical expansion B_k = sum_{i=0}^k S(k,i)(-1)^i i!/(i+1).
Used in Corollary 3 to rewrite Bernoulli numbers.

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Cite this review

Pith. "Pith review of A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind." pith.science (2026). https://pith.science/paper/VNVXLRIT

@misc{pith2026250522819,
  author       = {Pith},
  title        = {Pith review of: A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNVXLRIT}},
  note         = {Machine review of arXiv:2505.22819}
}

abstract

We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \[ \sum_{k=0}^{n} 2^{\,n-k}\,s(n,k)\,B_k \;=\; \sum_{k=0}^{n} b(n,k)\,\frac{(-1)^k\,k!}{k+1}. \] This parallels the classical Stirling--Bernoulli relation \[ B_n = \sum_{k=0}^{n} S(n,k)\,\frac{(-1)^k\,k!}{k+1}, \] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers.

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Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

[3]

Stenlund, On the Connection Between Stirling Numbers and Bessel Numbers,Elec- tron

D. Stenlund, On the Connection Between Stirling Numbers and Bessel Numbers,Elec- tron. J. Combin.29(1):P1.40, 2022. 4 2020Mathematics Subject Classification: Primary 11B68, 05A18, 11B73, 33C10. Keywords:Bernoulli number, Stirling number, Bessel number of first kind, operational iden- tity. (Concerned with sequences A008275 , A008277, A122850.) Received ; ...

work page 2022
[2]

Han and S

H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers,European J. Combin.29(2008), 1544–1554

work page 2008
[1]

Comtet,Advanced Combinatorics, Reidel, 1974

L. Comtet,Advanced Combinatorics, Reidel, 1974

1974

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

Stenlund, On the Connection Between Stirling Numbers and Bessel Numbers,Elec- tron

D. Stenlund, On the Connection Between Stirling Numbers and Bessel Numbers,Elec- tron. J. Combin.29(1):P1.40, 2022. 4 2020Mathematics Subject Classification: Primary 11B68, 05A18, 11B73, 33C10. Keywords:Bernoulli number, Stirling number, Bessel number of first kind, operational iden- tity. (Concerned with sequences A008275 , A008277, A122850.) Received ; ...

work page 2022

[2] [2]

Han and S

H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers,European J. Combin.29(2008), 1544–1554

work page 2008

[3] [1]

Comtet,Advanced Combinatorics, Reidel, 1974

L. Comtet,Advanced Combinatorics, Reidel, 1974

1974