A note on new type degenerate Srirling numbers of the first kind

Dae San Kim; Dmitry V. Dolgy; Kyo-Shin Hwang; Taekyun Kim

REVIEW 1 major objections 4 minor 12 references

A note on new type degenerate Srirling numbers of the first kind

T0 review · 1 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A one-parameter deformation of Stirling numbers of the first kind is shown to satisfy a finite-sum probabilistic identity that reduces to the classical random-variable formula when the parameter vanishes.

desk verdict The first-kind half is sound and the probabilistic identity is a clean Adell-Lekuona analog, but the second-kind definition (17) is inconsistent with the expansion and theorem that follow it. read the letter →

arxiv 2509.03415 v1 pith:JWC6C6KK submitted 2025-09-03 math.NT

Taekyun Kim , Dae san Kim , Kyo-Shin Hwang , Dmitry V. Dolgy This is my paper · ORCID

classification math.NT MSC 11B7311B8360-08

keywords newtypedegenerateStirlingnumbersofthefirstkindunsignedsecondprobabilisticrepresentationfinite-sumidentitiesexponentialexpectedvalues

topics The Collatz Conjecture

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 defines λ-deformed versions of the unsigned Stirling numbers of the first kind, called new type degenerate Stirling numbers of the first kind, and proves a closed finite-sum expression for them. The expression writes each number as a sum over m of powers of λ, binomial coefficients, ordinary signed Stirling numbers, and expected values of a degenerate power of a random sum S_k built from k independent products of a uniform(0,1) and an exponential(1) variable. This is a probabilistic representation in the same spirit as the classical identity expressing ordinary unsigned Stirling numbers as binom(n,k) times E[S_k^{n-k}], and it reduces to that identity as λ→0. The paper also defines, by inversion, new type degenerate Stirling numbers of the second kind and gives their finite-sum expression in terms of ordinary Stirling numbers of the second kind. If correct, the construction gives a coherent one-parameter family of combinatorial numbers that interpolate the classical Stirling numbers and are computable directly from them.

What carries the argument

The central object is the random sum S_k = U_1X_1 + ⋯ + U_kX_k and the identity E[e^{S_k}_λ(t)] = (1/((1/λ)log(1+λt)))^k log^k(1/(1−(1/λ)log(1+λt))). This identity links the generating function of the new first-kind numbers to an expectation, allowing the degenerate exponential to be expanded through the falling factorial (S_k)_{l,λ} and the logarithmic factor through ordinary Stirling numbers S1(m,k). That expansion is what produces the finite-sum probabilistic formula. For the second-kind family, the analogous engine is the expansion of (1/λ)(e^{λ(e^t−1)}−1) in powers of (e^t−1), which turns the defining series into a convolution of ordinary Stirling numbers of the second kind.

What would settle it

Expand Eq. (17) literally at k=1: the coefficient of t^1/1! on the left is (1/λ)·(d/dt)e_λ(e^t−1)|_{t=0}=1/λ, whereas Theorem 3.1 gives {1 1}^*_λ=1. Replacing the base with e^{λ(e^t−1)} makes the coefficient 1, matching the theorem and the ordinary λ→0 limit. Checking this first coefficient decides which definition the paper's result requires.

Watch

Extended reading notes

Core claim

Starting from the generating function for the unsigned new type degenerate Stirling numbers of the first kind, the paper proves Theorem 2.1: these numbers equal Σ_{m=k}^n λ^{n−m} binom(n,m) [m k]. The main result, Theorem 2.2, states that for n≥k≥1, [n k]^*_{−λ} = Σ_{m=k}^n λ^{m−k} binom(n,m) S1(m,k) E[(S_k)_{n−m,λ}], where S_k is the sum of k independent products of a uniform(0,1) and an exponential(1) random variable. The proof uses the identity E[e^{S_k}_λ(t)] = ((1/λ)log(1+λt))^{-k} log^k(1/(1−(1/λ)log(1+λt))), which converts the defining generating function into an expectation of a degenerate exponential. Letting λ→0 recovers the classical identity [n k] = binom(n,k) E[S_k^{n−k}]. The c

Load-bearing premise

The proof of the second-kind formula assumes that the defining generating function really uses e^{λ(e^t−1)} rather than the degenerate exponential e_λ(e^t−1) shown in the manuscript; if the printed expression is taken literally, the expansion in the proof is wrong.

Editorial extensions

If this is right

Each new first-kind number is a polynomial in λ with integer coefficients, thanks to the finite sum in Theorem 2.1, and the paper's tables show that a wide range of values can be computed directly from ordinary Stirling numbers.
Theorem 2.2 provides a finite-sum identity, so the expectation E[(S_k)_{n−m,λ}] can be replaced by the explicit k-fold integral displayed in Theorem 2.3, giving an integral representation for every new first-kind number.
Both new families limit to the ordinary Stirling numbers as λ→0, so the construction is a genuine one-parameter extension rather than an unrelated sequence.
The convolution structure in Theorems 2.1 and 3.1 means the new numbers are computable from binomial coefficients and classical Stirling numbers with no need to evaluate expectations numerically.
The generating-function identities give a route to recurrences and further identities for the new families by standard manipulations of exponential generating functions.

Reading between the lines

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

The printed definition (17), read literally with the degenerate exponential e_λ(e^t−1), does not support the expansion used to prove Theorem 3.1; the theorem goes through only if the intended base is e^{λ(e^t−1)}, and the λ→0 limit confirms that this is almost certainly the intended definition.
The same probabilistic device could define degenerate analogues of Bell, Dowling, or derangement numbers by substituting e^{λ(e^t−1)}-type bases into their generating functions, with the same ordinary-Stirling expansion producing finite-sum formulas.
Because S_k is a sum of k independent products of uniform and exponential variables, Theorem 2.2 suggests asymptotic and moment formulas for the new numbers as n grows, a direction the paper does not pursue.
The explicit polynomial-in-λ formulas in Theorems 2.1 and 3.1 invite further study of divisibility properties or roots as λ varies, which is not addressed in the paper.

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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

Quick read of arXiv:2509.03415. First the headline: the first-kind section is correct and worth a look; the second-kind section has a real inconsistency in its defining equation, and the paper as printed doesn't support Theorem 3.1.

The genuinely new object is the sequence [n k]*_λ defined by (7)-(8). Theorem 2.1 gives a convolution formula in ordinary Stirling numbers, and Theorem 2.2 gives a probabilistic identity that degenerates cleanly to Adell-Lekuona. The proofs are short but correct — the Cauchy product in (14) is fine, and the λ→0 recovery checks out. That's a legitimate, if modest, contribution to the degenerate-number program.

The soft spot is Eq (17). As typeset, the generating function is (1/k!)[(e_λ(e^t-1)-1)/λ]^k, with the degenerate exponential e_λ. But the very next display, (18), expands it as if the base were (e^{λ(e^t-1)}-1)/λ: the Stirling expansion used there is for ordinary e^z, not e_λ(z). These are not the same function, and the difference is load-bearing. For k=1, the printed (17) has a t^2/2! coefficient of 1/λ - 1/2, whereas Theorem 3.1 and the table require 1+λ. So as printed, the λ→0 limit to ordinary Stirling numbers of the second kind fails, and Theorem 3.1 is false. The intended base is almost certainly e^{λ(e^t-1)}, which gives the expansion in (18) and the stated values. The fix is one character, but the authors need to make it and re-verify the table.

Beyond that, the paper is honest and clearly written, though thin: no applications, no open problems, and the new families are variants of the authors' existing degenerate machinery. The self-citations are mostly to that same line, but the derivations are transparent.

Bottom line: the first-kind half is publishable, and the second-kind half is a typo away from being publishable. I'd send it to a referee, with a note asking the authors to reconcile (17) with (18) and the table. It's not a paper that will change anyone's practice, but it's not a waste of a referee's time either.

Referee Report

1 major / 4 minor

Summary. The paper defines a one-parameter family of 'new type degenerate' unsigned Stirling numbers of the first kind via the generating function (8), proves an explicit finite convolution formula (Theorem 2.1), and derives a finite-sum probabilistic representation (Theorem 2.2) in the style of Adell-Lekuona, with the parameter limit recovering the classical identity (5). It then introduces, by inversion, a 'new type degenerate' Stirling number of the second kind and claims an analogous convolution formula (Theorem 3.1). The first-kind results are derived cleanly; the second-kind section, as printed, contains a mismatch between the definition and the proof.

Significance. The first-kind representation (15) is correct and gives a genuine degenerate analogue of the Adell-Lekuona identity, with a transparent derivation from a probabilistic calculation. If the typesetting error in (17) is corrected, Theorem 3.1 provides a natural inverse family and explicit convolution formula. The results are modest—the numbers are finite sums of classical Stirling numbers—but they are coherent and the probabilistic interpretation is a useful addition. The paper's proofs are elementary and checkable; no parameter fitting or circular reasoning is involved.

major comments (1)

[Sec. 3, Eq. (17) through Theorem 3.1] As printed, Eq. (17) defines the second-kind numbers using e_λ(e^t−1) = (1+λ(e^t−1))^{1/λ}. This is inconsistent with the expansion in Eq. (18), which is valid for e^{λ(e^t−1)}. For example, for k=1 the coefficient of t^2/2! in the printed (17) is 2/λ−1, while Theorem 3.1 and the table give {2 1}^*_λ = 1+λ. The stated λ→0 limit also fails for the printed definition. Since Theorem 3.1 and its table depend entirely on Eq. (18), the theorem is unproved as stated. The intended definition is evidently ((e^{λ(e^t−1)}−1)/λ)^k/k!, which is the compositional inverse of the first-kind generating function; please correct (17) and the note following it.

minor comments (4)

[Title and abstract] Typo in title: 'Srirling' should be 'Stirling'; the header also has an unwanted space in 'DEGENERA TE'.
[Sec. 3, after Theorem 3.1] The sentence 'we illustrate the values of the unsigned new type degenerate Stirling numbers of the first kind' should refer to the second-kind numbers defined in (17).
[Eq. (15) paragraph] 'By taking λ → 0 in (15) and using (9)' is ungrammatical; suggest 'Taking λ → 0 in (15) and using (9), we recover (5).'
[Throughout] Minor typos: 'plynomial' should be 'polynomial', 'Furthemore' should be 'Furthermore', 'probabilty' should be 'probability'. Some displayed tables lack column headers, which would help readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new-number identities are direct generating-function expansions; the self-citations are not load-bearing.

full rationale

The central results are obtained by direct coefficient extraction from the stated generating functions, not by fitting or by assuming the target. Theorem 2.1 (Eq. 10) expands (8) using the standard series for log(1−λt) and the definition of ordinary Stirling numbers of the first kind; Theorem 2.2 (Eqs. 11–15) computes the degenerate moment generating function of Sk explicitly, multiplies by the reciprocal factor, and compares coefficients; Theorem 3.1 (Eq. 18) expands the second-kind generating function using the ordinary Stirling expansion of (e^{λ(e^t−1)}−1)^k. These are algebraic derivations from the definitions. The citations to the authors' earlier papers [4,5,6,7,8,12] provide background definitions and standard identities; the definitions used here are restated in the paper and are independently verifiable, so they are not load-bearing self-citations. No parameter is fitted to reproduce a target, and no uniqueness claim is imported from the authors' prior work. The only notable issue is a non-circular correctness/typo concern: Eq. (17) prints e_λ(e^t−1), while the expansion in Eq. (18) is valid for the compositional inverse e^{λ(e^t−1)}; this is an algebraic inconsistency, not a circularity. Hence score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

No constants are fitted to data; lambda is a free deformation parameter, not a fitted number. The paper relies on standard generating function facts, a standard probabilistic model, and its own definitions of the new sequences.

assumptions (4)

standard math Classical Stirling number generating function identities in (2), (4), (10), and (18).
Used throughout as the starting expansions for the new families; standard results from Comtet [3] and Roman [9].
standard math Degenerate exponential e_lambda(t) = (1+lambda t)^{1/lambda} and its product property e_{x+y} = e_x e_y.
Needed for the probabilistic factorization in (12) and for the definitions of degenerate Stirling numbers; follows directly from the definition but is not proved in the note.
domain assumption Probabilistic model with U independent uniform on (0,1) and X independent exponential with rate 1, plus interchange of expectation and coefficient extraction.
Underlies the whole probabilistic representation (11)-(14); the model is standard but is an external modeling assumption.
ad hoc to paper The definitions (7) and (17) determine well-defined formal power series and hence well-defined number sequences.
These generating functions are the paper's own construction of the new sequences; they are not established background and their validity is the content of the paper.

invented entities (2)

Unsigned new type degenerate Stirling numbers of the first kind [n k]*_lambda
purpose: Main object of study; generalizes ordinary unsigned Stirling numbers with a deformation parameter lambda.
Defined by the generating functions (7)-(8); no external falsifiable handle, only internal consistency with the classical lambda to 0 limit.
New type degenerate Stirling numbers of the second kind {n k}^*_lambda
purpose: Inversion pair for the first kind; defined by (17).
Defined by a generating function with no external falsifiable handle; as printed in (17) it is internally inconsistent with its own expansion (18).

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Cite this review

Pith. "Pith review of A note on new type degenerate Srirling numbers of the first kind." pith.science (2026). https://pith.science/paper/JWC6C6KK

@misc{pith2026250903415,
  author       = {Pith},
  title        = {Pith review of: A note on new type degenerate Srirling numbers of the first kind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWC6C6KK}},
  note         = {Machine review of arXiv:2509.03415}
}

read the original abstract

We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random variables, we express our new numbers as finite sums of multiples of the expectations of certain random variables. We also provide a representation of these new numbers as finite sums involving the classical unsigned Stirling numbers of the first kind. As an inversion formula, we define a corresponding sequence of new type degenerate Stirling numbers of the second kind. We derive expressions for these numbers as finite sums that involve the Stirling numbers of the second kind.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

A.; Lekuona, A

Adell, J. A.; Lekuona, A. Closed form expressions for the Stirling numbers of the first kind, Integers 17 (2017), Paper No. A26, 4 pp

work page 2017
[2]

S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications, Proc

Aydin, M. S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications, Proc. Jangjeon Math. Soc. 25 (2022), no. 2, 195-203

work page 2022
[3]

Advanced combinatorics

Comtet, L. Advanced combinatorics. The art of finite and infinite expansions, Revised and en- larged edition, D. Reidel Publishing Co., Dordrecht, 1974

work page 1974
[4]

S.; Kim, T

Kim, D. S.; Kim, T. Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys. 31 (2024), no. 4, 682-690

work page 2024
[5]

Kim, T.; Kim, D. S. Spivey-type recurrence relations for degenerate Bell and Dowling polyno- mials, Russ. J. Math. Phys. 32 (2025), no. 2, 288-296

work page 2025
[6]

Kim, T.; Kim, D. S. Some identities on degenerate r-Stirling numbers via boson operators,Russ. J. Math. Phys. 29 (2022), no. 4, 508-517

work page 2022
[7]

S.; Dolgy, D

Kim, T.; Kim, D. S.; Dolgy, D. V . Probabilistic degenerate derangement polynomials, Math. Comput. Model. Dyn. Syst. 31 (2025), no. 1, 2529188

work page 2025
[8]

S.; Xu, R

Ma, Y .; Kim, T.; Kim, D. S.; Xu, R. Recurrence for probabilistic extension of Dowling polyno- mials, Open Math. 23 (2025), no. 1, Paper No. 20250150, 7 pp

work page 2025

Show all 12 references

[9]

The umbral calculus, Pure and Applied Mathematics 111, Academic Press, Inc

Roman, S. The umbral calculus, Pure and Applied Mathematics 111, Academic Press, Inc. [Har- court Brace Jovanovich, Publishers], New York, 1984

1984
[10]

Introduction to probability models,Thirteenth edition, Academic Press, Lon- don, 2024

Ross, Sheldon M. Introduction to probability models,Thirteenth edition, Academic Press, Lon- don, 2024

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Identities and relations related to combinatorial numbers and polynomials, Proc

Simsek, Y . Identities and relations related to combinatorial numbers and polynomials, Proc. Jangjeon Math. Soc. 20 (2017), no. 1, 127-135

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[12]

S.; Zhang, W.Probabilistic degenerate poly-Bell polynomials associated with random variables,Math

Xue, P.; Ma, Y .; Kim, T.; Kim, D. S.; Zhang, W.Probabilistic degenerate poly-Bell polynomials associated with random variables,Math. Comput. Model. Dyn. Syst.31 (2025), no. 1, Paper No. 2497367, 17 pp. DEPARTMENT OF MATHEMATICS , K WANGWOON UNIVERSITY , S EOUL 139-701, R EPUB...

2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

A.; Lekuona, A

Adell, J. A.; Lekuona, A. Closed form expressions for the Stirling numbers of the first kind, Integers 17 (2017), Paper No. A26, 4 pp

work page 2017

[2] [2]

S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications, Proc

Aydin, M. S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications, Proc. Jangjeon Math. Soc. 25 (2022), no. 2, 195-203

work page 2022

[3] [3]

Advanced combinatorics

Comtet, L. Advanced combinatorics. The art of finite and infinite expansions, Revised and en- larged edition, D. Reidel Publishing Co., Dordrecht, 1974

work page 1974

[4] [4]

S.; Kim, T

Kim, D. S.; Kim, T. Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys. 31 (2024), no. 4, 682-690

work page 2024

[5] [5]

Kim, T.; Kim, D. S. Spivey-type recurrence relations for degenerate Bell and Dowling polyno- mials, Russ. J. Math. Phys. 32 (2025), no. 2, 288-296

work page 2025

[6] [6]

Kim, T.; Kim, D. S. Some identities on degenerate r-Stirling numbers via boson operators,Russ. J. Math. Phys. 29 (2022), no. 4, 508-517

work page 2022

[7] [7]

S.; Dolgy, D

Kim, T.; Kim, D. S.; Dolgy, D. V . Probabilistic degenerate derangement polynomials, Math. Comput. Model. Dyn. Syst. 31 (2025), no. 1, 2529188

work page 2025

[8] [8]

S.; Xu, R

Ma, Y .; Kim, T.; Kim, D. S.; Xu, R. Recurrence for probabilistic extension of Dowling polyno- mials, Open Math. 23 (2025), no. 1, Paper No. 20250150, 7 pp

work page 2025