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Some identities of type 2 Degenerate Bernoulli polynomials of the second kind

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Pith's one-line read This paper introduces type 2 degenerate Bernoulli polynomials of the second kind and derives four identities linking them to degenerate Stirling and central factorial numbers.

desk verdict Routine degenerate-polynomial paper whose headline identity is false; a simple counterexample kills Theorem 2.1's second half. read the letter →

arxiv 1908.06587 v1 pith:OJZ6X7VY submitted 2019-08-19 math.NT

Taekyun Kim , Lee-Chae jang , Dae San Kim , Han-Young Kim This is my paper · ORCID
classification math.NT MSC 11B6811B7305A19
keywords type2degenerateBernoullipolynomialsofthesecondkindStirlingnumbersfirstcentralfactorialexponentialgeneratingfunctions
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 introduces a degenerate analogue of the type 2 Bernoulli polynomials of the second kind, written $b^*_{n,\lambda}(x)$, together with higher-order analogues $b^{*,(k)}_{n,\lambda}(x)$, and derives four families of identities connecting them to degenerate Bernoulli polynomials of the second kind, degenerate Stirling numbers of the first and second kind, and degenerate central factorial numbers. The guiding idea is to replace ordinary exponentials with the degenerate exponential $(1+\lambda t)^{1/\lambda}$ so that classical Bernoulli--Stirling identities become one-parameter families that reduce to the classical formulas as $\lambda\to 0$. If the identities are correct, they give finite closed-form evaluations of weighted sums in which these polynomial families appear, including the compact special case in Theorem 2.2.

What carries the argument

The machinery is the degenerate exponential $e_\lambda(t)=(1+\lambda t)^{1/\lambda}$ and its compositional inverse $\log_\lambda t=(t^\lambda-1)/\lambda$, used throughout to define degenerate versions of classical objects. The central identity-generating object is the defining series $((1+t)-(1+t)^{-1})/\log_\lambda(1+t)\,(1+t)^x$, whose coefficients are $b^*_{n,\lambda}(x)$, together with the analogous series for degenerate Bernoulli polynomials of the second kind, degenerate Stirling numbers of the first and second kind, and degenerate central factorial numbers. The proofs work by substituting a composed argument, such as $e^{2t}-1$ or $\log_\lambda(1+t)$, into one of these series, expanding the result in two different ways, and equating coefficients of $t^n/n!$.

What would settle it

Evaluate equation (23) at $n=1$, $r=2$, $\lambda=0$, $x=0$: the left-hand side equals $1$ and the right-hand side equals $2$, so the claimed identity fails at that point. Computing both sides for small $n$ and $r>1$ with $\lambda=0$ settles the theorem.

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Extended reading notes

Core claim

The central discovery is a set of generating-function identities for the newly defined type 2 degenerate Bernoulli polynomials of the second kind and their order-$k$ analogues. Theorem 2.1 writes $b^*_{n,\lambda}(x)$ as $b^{(1)}_{n,\lambda}(x)+b^{(1)}_{n,\lambda}(x-1)$ and gives a convolution identity for the order-$r$ degenerate Bernoulli polynomials of the second kind with Stirling numbers of the second kind. Theorem 2.2 evaluates a weighted sum of order-$k$ type 2 degenerate Bernoulli polynomials with degenerate Stirling numbers of the second kind, and the special case $x=k$ produces the compact formula $2^{n+k}S_{2,\lambda/2}(n+k,k)=\binom{n+k}{k}\sum_{l=0}^n b^{*,(k)}_{l,\lambda}(k)S_{2,\lambda}(n,l)$. Theorem 2.3 expands the order-$k$ polynomials as a finite sum of negative-order type 2 degenerate Bernoulli polynomials times degenerate Stirling numbers of the first kind. Theorem 2.4 links degenerate central factorial numbers of the second kind, degenerate Stirling numbers of the first kind, and order-$k$ degenerate Bernoulli polynomials of the second kind. All four proofs compare two expansions of the same generating function after a substitution such as $t\mapsto e^{2t}-1$ or $t\mapsto \log_\lambda(1+t)$.

Load-bearing premise

The derivation of the second identity in Theorem 2.1 assumes that raising the type 2 Bernoulli generating function to the $r$-th power can be expanded through first-order type 2 Bernoulli polynomials; this substitution is only valid when $r=1$.

Editorial extensions

If this is right

  • The order-$r$ degenerate Bernoulli polynomials of the second kind satisfy an explicit finite convolution with Stirling numbers of the second kind, so the sum $\sum_{m=0}^n b^{(r)}_{m,\lambda}(x)S_2(n,m)$ has a closed form.
  • The order-$k$ type 2 degenerate Bernoulli polynomials can be expanded in terms of negative-order type 2 degenerate Bernoulli polynomials with degenerate Stirling numbers of the first kind, giving an inversion-type relation between the two families.
  • Setting $\lambda\to 0$ in each identity should recover the corresponding nondegenerate identity for type 2 Bernoulli polynomials of the second kind, so the paper supplies one-parameter lifts of classical relations.
  • The special case $x=k$ in Theorem 2.2 gives the compact evaluation $2^{n+k}S_{2,\lambda/2}(n+k,k)=\binom{n+k}{k}\sum_{l=0}^n b^{*,(k)}_{l,\lambda}(k)S_{2,\lambda}(n,l)$.

Reading between the lines

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

  • A natural next step would be to invert the Stirling transform in the Theorem 2.1 convolution to solve directly for $b^{(r)}_{n,\lambda}(x)$; the paper does not write this inversion.
  • The same composition technique could be applied to type 2 degenerate Euler polynomials of the second kind, yielding analogous identities with degenerate Euler numbers.
  • Because Theorem 2.4 mixes degenerate central factorial numbers with degenerate Stirling numbers, comparing the two sides at small $n$ could reveal a cleaner combinatorial interpretation of the coefficients when viewed as set partitions with two different statistics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper defines type 2 degenerate Bernoulli polynomials of the second kind b*_{n,λ}(x) via the generating function (24), and their higher-order analogues b*^{(α)}_{n,λ}(x) via (28). It then derives four theorems: Theorem 2.1 gives two identities, the first relating b*_{n,λ}(x) to the degenerate Bernoulli polynomials of the second kind of order 1, and the second (Eq. 23) relating the higher-order b^{(r)}_{m,λ}(x) to type 2 Bernoulli polynomials and Stirling numbers of the second kind. Theorem 2.2 gives an identity involving degenerate Stirling numbers of the second kind, Theorem 2.3 expresses the higher-order polynomials in terms of negative-order type 2 degenerate Bernoulli polynomials and degenerate Stirling numbers of the first kind, and Theorem 2.4 gives an identity involving degenerate central factorial numbers, degenerate Stirling numbers, and higher-order degenerate Bernoulli polynomials of the second kind. The proofs are formal manipulations of generating functions.

Significance. If the identities were correct, they would add useful new relations to the catalog of degenerate special polynomials, and the paper introduces previously unnamed objects (type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues). However, the second identity of Theorem 2.1 is false, and because this identity is explicitly advertised in the introduction and conclusions as a main contribution, the paper's central claim does not stand as written. The first identity of Theorem 2.1 and Theorems 2.2–2.4 appear to be derived correctly, so the paper could in principle be revised by removing the false identity, but as submitted the main result is unsound.

major comments (1)
  1. [Section 2, Eq. (21) and Theorem 2.1 (Eq. (23))] The expansion in Eq. (21) of (λt/(e^{λt}-e^{-λt}))^r e^{(2x-λr)t} in terms of the first-order type 2 Bernoulli polynomials B*_k is invalid for r>1. The generating function of B*_k is the first power z/(e^z-e^{-z}) e^{yz}; raising it to the r-th power yields the higher-order type 2 Bernoulli polynomials of order r, not the same first-order polynomials. Consequently Eq. (21) is false, and the second identity of Theorem 2.1, Eq. (23), is false. A concrete counterexample is n=0, r=2, x=0, any λ: the left side of (23) is b^{(2)}_{0,λ}(0)S2(0,0)=1, while the right side is B*_0(-2) S2(2,2)/(2 choose 2) 2^2 = 4. Thus Eq. (23) asserts 1=4. The first identity of Theorem 2.1 is not affected, but the second is a headline claim and the error is structural, not a typographical slip.
minor comments (4)
  1. [Abstract] The abstract contains multiple spelling and grammatical errors ('I recent years', 'dege nerate', 'hihger', 'spe- cial'), and the phrase 'type 2 degenerate Bernoulli polynomials of the second' should include 'kind' at the end of the sentence.
  2. [Eq. (34)] In Eq. (34), the term β^{*(-k)}_{l,λ} S1,λ(n.l) should read β^{*(-k)}_{l,λ}(x) S1,λ(n,l); the variable x is missing and the period in 'n.l' should be a comma.
  3. [Eq. (33) and notation] The notation e^{-1}_λ(t) in Eq. (33) is ambiguous; it should be defined explicitly as the reciprocal of e_λ(t) or as e_λ^{-1}(t)=(1+λt)^{-1/λ}, to avoid confusion with an inverse function.
  4. [Throughout] There are numerous typographical errors and inconsistent notations (e.g., 'n.l' in Eq. (34), missing commas in sums, and inconsistent use of 'kind' vs 'kin d'); the manuscript needs careful proofreading.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the identities are derived directly from generating-function definitions; the main flaw in Eq. (21) is a mathematical error, not circular reasoning.

full rationale

I walked the paper's derivation chain. The central objects are introduced by explicit generating functions, and the identities in Theorems 2.1-2.4 are obtained by standard formal-manipulation steps: factoring the generating function, substituting t -> e_λ(t)-1 or t -> log_λ(1+t), expanding in known Stirling-number series, and comparing coefficients. No fitted parameter is introduced and no target identity is assumed as an input. The relation b*_{n,λ}(x) = b^{(1)}_{n,λ}(x) + b^{(1)}_{n,λ}(x-1) is immediate from the defining generating functions, since (1+t)-(1+t)^{-1} = t + t/(1+t); this is a direct consequence of definitions, not a circular prediction. The self-citations that appear are to standard definitions of type 2 Bernoulli polynomials, degenerate Stirling numbers, and degenerate central factorial numbers; those citations are not load-bearing for the new identities, and the definitions are parameter-free and independent of the target results. The paper's serious defect is mathematical rather than circular: Eq. (21) incorrectly replaces the r-th power of the type 2 Bernoulli generating function by the first-order type 2 Bernoulli generating function, which makes the second identity in Theorem 2.1 false (for n=1, r=2, λ=0, x=0 the left side is 1 and the right side is 2). That is an invalid substitution, not an assumption of the conclusion, so it does not constitute circularity. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper's central content is definitional; no parameters are fitted to data. The main assumptions are standard formal power series operations and the correctness of cited generating functions. The invented polynomials have no independent existence outside the paper; their value is purely definitional.

assumptions (3)
  • standard math Formal power series operations (substitution, composition, coefficient extraction) are valid.
    Used throughout Sections 2 and 3 without comment.
  • domain assumption The degenerate logarithm log_λ(t) = (t^λ - 1)/λ is the compositional inverse of e_λ(t) = (1+λt)^{1/λ}.
    Definition in (17); used in substitutions in (21), (30), (34), (36).
  • standard math The cited generating functions for Stirling numbers (10), (11), degenerate Stirling numbers (31), (35), and central factorial numbers (16) are correct.
    Taken from prior literature; the paper depends on them for coefficient expansions.
invented entities (2)
  • Type 2 degenerate Bernoulli polynomials of the second kind b*_{n,λ}(x)
    purpose: New polynomial family defined in (24); the paper derives identities for it.
    No external applications or predictions; the entity exists only through its generating function definition.
  • Higher-order analogues b*^{(α)}_{n,λ}(x)
    purpose: Defined in (28) to generalize the family; used in Theorems 2.2 and 2.3.
    Same as above; no independent evidence beyond the defining series.

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Pith. "Pith review of Some identities of type 2 Degenerate Bernoulli polynomials of the second kind." pith.science (2026). https://pith.science/paper/OJZ6X7VY

@misc{pith2026190806587,
  author       = {Pith},
  title        = {Pith review of: Some identities of type 2 Degenerate Bernoulli polynomials of the second kind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJZ6X7VY}},
  note         = {Machine review of arXiv:1908.06587}
}
read the original abstract

I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specially, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second kind and degenerate Bernoulli polynomials of the second kind, and identity involving hihger-order analogues of those polynomials and the degenerate stirling number of the second kin, and and expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate stirling number of the first kind.

Discussion (0). Continue with ORCID to comment.

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