REVIEW 3 major objections 4 minor 4 references
Some identities which involve Stirling numbers
T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This note reports exact identities that rewrite Stirling numbers of the first and second kind as sums over integer partitions, with the second-kind forms passing through complete Bell polynomials and the Lambert W function.
desk verdict Eq. (2.1) is stated over unordered partitions but only works when read as ordered compositions; once fixed, the note is a harmless, useful set of Stirling identities. 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 central object is the $q+1$-fold weighted binomial sum $c_l(n,q;\{n_i\}) = \sum_{\{j_i\}\in\Pi(l,q+1)} \prod_{i=0}^q (i+1)^{n_i-j_i}\binom{n_i}{j_i}$, together with its averaged version $\bar c_l(n,q;w)$ in which the $n_i$ are summed over ordered partitions of $n$ into $q+1$ parts with a weight that is either 1 or a product of rising factorials. The closed forms come from encoding the single-rising-factorial case in the generating function $\hat C(t;q,m,\alpha) = [(1-(\alpha+1)t)^m\prod_{i=0}^q(1-(i+1)t)]^{-1}$; expanding its logarithm and applying complete exponential Bell polynomials, the polynomials that organise derivatives of a composite function, converts the nested partition sums into single sums or closed forms. The Lambert-W form enters through the exponential generating function of the Sidi-polynomial coefficients that appear in the Bell-polynomial identities.
What would settle it
Expand the right-hand side of Eq. (2.1) for q=3, p=4 using the five unordered partitions of 4; the total will not be 1624, whereas the paper's own Eq. (2.3) reaches 1624 only by using seven ordered compositions. This one arithmetic check settles whether Eq. (2.1) is valid as written.
Extended reading notes
Core claim
The paper's claim is that the unsigned Stirling numbers of the first kind can be written as a sum over ordered partition contributions, with the signed version following by the standard parity factor, and that the weighted partition sums defined in the paper collapse into closed forms. Equation (3.9) equates the basic weighted sum with a binomial coefficient times a Stirling number of the second kind; Eqs. (3.11)-(3.12) handle the single-rising-factorial case as a single sum; Eq. (3.13) extends this to several rising factorials by cyclic symmetrisation; Eq. (3.26) gives a finite-sum expression for the coefficient $\hat c(j,q,m,\alpha)$; Eq. (3.30) expresses Stirling numbers of the second kind through complete Bell polynomials evaluated at power sums; and Eq. (3.36) gives a Lambert-W generating-function form. In the author's own telling these are useful formulas encountered along the way, not a claim to new mathematics.
Load-bearing premise
The first identity only works if the sum over 'partitions' is read as ordered lists, even though the paper's defining equation says the symbol means unordered collections.
Editorial extensions
If this is right
- Equation (2.1) gives unsigned Stirling numbers of the first kind as an all-positive sum over partitions of an auxiliary integer, with signed values obtained by the standard parity factor.
- Equation (3.9) collapses the original double partition sum to a binomial coefficient times a Stirling number of the second kind.
- Equations (3.11)-(3.13) reduce weighted sums carrying one or several rising factorials to single-sum expressions, with several factorials handled by cyclic symmetrisation.
- Equation (3.26) replaces the $n$-dependent upper limit of the earlier single sum by a finite sum, so the large-$n$ behaviour of these quantities can be studied without infinite sums.
- Equation (3.30) identifies Stirling numbers of the second kind with complete Bell polynomials evaluated on power sums, and Eq. (3.36) gives a Lambert-W generating-function variant.
Reading between the lines
- As an extension, the all-positive form of the first-kind sum could make Eq. (2.1) numerically stable for large parameters, where alternating Stirling-recurrence evaluations suffer cancellation; the note does not test this.
- The factorised generating function behind Eq. (3.20) suggests that replacing the weights $(i+1)^{n_i-j_i}$ by other geometric factors should yield analogous Bell-polynomial identities for sequences beyond Stirling numbers; that generalisation is not in the note.
- In the small-$x$ context that motivated the note, Eq. (3.26) turns an asymptotic problem into a finite computation, so leading large-$n$ behaviour of the weighted partition sums can be read off term by term; the note leaves that extraction to the companion small-$x$ work.
- A reader who wants to know whether Eq. (3.36) is practically useful could compare derivative-at-the-origin evaluation against standard Stirling recurrences for large $n$ and $k$; the note reports no such comparison.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript is a short note reporting identities that express Stirling numbers of the first and second kinds, together with binomial coefficients, in terms of sums over integer partitions and complete Bell polynomials. The central results are Eq. (2.1) for unsigned Stirling numbers of the first kind; Eqs. (3.9), (3.11), (3.13), and (3.26) for weighted sums over ordered partitions with rising-factorial weights; and Eqs. (3.30) and (3.36), which relate Stirling numbers of the second kind to Bell polynomials and to the coefficients of Sidi polynomials. The note is motivated by the small-$x$ behaviour of parton distribution functions and makes no claim of originality. Apart from a numerical example for Eq. (2.1), no derivations or machine checks are provided for the main identities.
Significance. Should the identities be correct, they provide explicit and apparently compact representations of Stirling numbers; in particular Eqs. (3.11)-(3.13) and (3.26) could support asymptotic analyses in $n$ for small-$x$ resummation. The author is transparent that the identities may already be known and does not fit any external benchmark, so there is no circularity. The concrete value in the paper is the generating-function method leading to Eq. (3.26). However, the note in its current form cannot be used as written because Eq. (2.1) is false under the stated definition of $\Pi(p)$, and because the main identities in Section 3 are asserted without derivations or verification code.
major comments (3)
- [Section 2, Eq. (2.1)] The sum in Eq. (2.1) is over $\Pi(p)$, defined in Eqs. (1.3)-(1.5) as unordered integer partitions with $1\le l_1\le\cdots\le l_{m_a}\le p$, but the summand depends on the order of the $l_m$ through the partial sums $\sum_{k=1}^m(l_k+1)$. The footnote on page 1 explicitly requires that the conventional ordering of an unordered partition must not affect the result. For $p=4$, $q=3$, the unordered partitions give $504+420+280+90=1294$, not $\binom{7}{3}=1624$; the value 1624 is obtained only when the sum is interpreted as running over ordered positive compositions with at most $q$ parts. The identity is therefore false as written. It can be repaired by replacing $\Pi(p)$ with the set of ordered positive compositions of length at most $q$, and the surrounding text, which already says 'ordered integer partitions', should be updated consistently.
- [Section 3, Eq. (3.13)] Equation (3.13) is a central identity for the case of $\lambda$ rising factorials, but it is introduced without derivation via 'one obtains', and the final cyclic-permutation instruction is not sufficient to reconstruct the general expression. The left-hand side involves $2(q+1)$ nested sums, while the right-hand side is a cyclically symmetrized sum with a Kronecker constraint and with summation ranges that are only partially specified. A derivation, or at least a fully written-out $\lambda=3$ case, should be supplied so that the identity can be checked. This is load-bearing because Eq. (3.13) is what connects the multiple-rising-factorial sums to the single-rising-factorial results of Eqs. (3.11) and (3.12).
- [Section 3, Eqs. (3.19)-(3.26)] The passage from the generating function $\hat C(t;q,m,\alpha)$ to the closed form in Eq. (3.26) is summarized as 'after some algebra'. This is a load-bearing step, since Eq. (3.26) is proposed as the finite-sum alternative suitable for the $n\to\infty$ limit. The preceding Bell-polynomial identity in Eq. (3.25) alone does not make Eq. (3.26) obvious, and the expression contains terms with $(i-\alpha)^m$ in the denominator that require care. Please provide the intermediate algebra or, failing that, a machine-checkable certificate such as a short computer-algebra script that verifies Eq. (3.26) for ranges of the parameters.
minor comments (4)
- [Section 1, Eq. (1.3)] The notation '$\Pi(p) , \Pi^{(p)}$' is confusing; the two sets should be denoted by clearly distinct symbols, such as $\Pi(p)$ and $\overline{\Pi}(p)$, consistently with the surrounding text.
- [Section 3, after Eq. (3.13)] The phrase '$(1 \to 2 \to 3 \to \ldots \lambda \to 1)^{\lambda-1}$ times' is too terse; a fully explicit version of the cyclic sum, at least for $\lambda=3$ and $\lambda=4$, would improve readability and verifiability.
- [Section 3, after Eq. (3.9)] The claim that the sequences $\{\bar c_l(n,q;1)\}_n$ and $\{\bar c_l(n,q;1)\}_q$ are not included in the OEIS for $l\ge1$ is not verifiable without OEIS query identifiers or accession numbers; please add them or soften the claim.
- [General] The manuscript would benefit from a reproducibility statement or supplementary code: because the identities are presented without proofs, a short script that evaluates the left- and right-hand sides of each displayed identity for random parameter values would substantially increase confidence in the results.
Circularity Check
No circular reasoning: the note derives explicit identities for Stirling numbers and binomial coefficients by direct algebraic manipulation, generating functions, and Bell polynomials, with no fitted inputs and no load-bearing self-citations.
full rationale
The paper is a self-contained set of algebraic derivations. Each identity is obtained from definitions (Stirling numbers, binomial coefficients, integer partitions) through generating-function or Bell-polynomial manipulations, and the conclusions are not assumed as inputs. No external data are fitted, and no parameter is renamed as a prediction. The only reference to the authors' own work is contextual (ref. [2], the ongoing small-x project), and it plays no role in proving the identities. The potential issue noted by the reader, that Eq. (2.1) is written as a sum over unordered partitions while its summand depends on the order of elements, is an internal consistency/correctness concern about the stated notation, not a circular-reasoning defect; it does not make the derivation circular. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Definitions of Stirling numbers of the first and second kind via factorial expansions and combinatorial interpretations.
- standard math Properties of complete exponential Bell polynomials, including the binomial-like summation formula used in Eq. (3.29).
- standard math Counting formula for ordered partitions with a fixed number of parts, Eq. (1.13).
Cite this review
Pith. "Pith review of Some identities which involve Stirling numbers." pith.science (2026). https://pith.science/paper/JQXVRFT6
@misc{pith2026250521197,
author = {Pith},
title = {Pith review of: Some identities which involve Stirling numbers},
year = {2026},
howpublished = {\url{https://pith.science/paper/JQXVRFT6}},
note = {Machine review of arXiv:2505.21197}
}
abstract
During the course of an ongoing work on the small-$x$ behaviour of parton distribution functions, some identities have been found which involve Stirling numbers of the first and the second kind, as well as binomial coefficients. Without any claim of originality I report them in this note.
Reference graph
Works this paper leans on
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[1]
" write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...
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[2]
J. Malenfant, Finite, closed-form expressions for the partition function and for euler, bernoulli, and stirling numbers, 2011
work page 2011
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[3]
M. Bonvini, S. Frixione and G. Stagnitto, Improved small- x resummation for DGLAP splitting functions: HELL 4.0 , http://arxiv.org/abs/25xx.yyyyy 25xx.yyyyy
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[4]
Sidi, Practical Extrapolation Methods: Theory and Applications
A. Sidi, Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2003
work page 2003
Reviewed August 7, 2026 · model on record in the stance chip above.
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