A note on degenerate Euler and Bernoulli polynomials of complex variable

Dae San Kim; Hyunseok Lee; Taekyun Kim

REVIEW 1 major objections 5 minor 1 cited by

A note on degenerate Euler and Bernoulli polynomials of complex variable

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Pith's one-line read Degenerate Euler and Bernoulli polynomials of a complex variable split into real and imaginary parts, yielding the degenerate cosine-Euler, sine-Euler, cosine-Bernoulli, and sine-Bernoulli families.

desk verdict A standard, correct special-functions note that adds degenerate cosine/sine Euler and Bernoulli polynomial families; the only real defect is a misprinted binomial coefficient in Theorem 2.7 that the proof itself corrects. read the letter →

arxiv 1908.03783 v1 pith:EGACCY3E submitted 2019-08-10 math.NT

Dae San Kim , Taekyun kim , Hyunseok Lee This is my paper · ORCID

classification math.NT MSC 11B6811B83

keywords degeneratecosine-Eulerpolynomialssine-Eulercosine-Bernoullisine-Bernoullicosine-polynomialssine-polynomialsStirlingnumbersofthesecondkind

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 establishes that the 'new type' Euler polynomials introduced in [15], together with their Bernoulli counterparts, have a natural degenerate ($\lambda$-parameter) version that comes directly from inserting a complex variable $x+iy$ into the degenerate Euler and Bernoulli polynomials defined in (4)-(5) and separating real from imaginary parts. This produces four families — degenerate cosine-Euler, sine-Euler, cosine-Bernoulli, and sine-Bernoulli polynomials — and the paper derives explicit finite-sum formulas for them in terms of ordinary and degenerate Stirling numbers, along with reflection and translation identities. The point of doing this is that the construction answers, affirmatively, a reviewer's question about whether the new type Euler polynomials can be obtained by considering Euler polynomials of complex variable and treating the real and imaginary parts separately. A sympathetic reader should care because the complex-variable viewpoint unifies these separately defined polynomial families and gives a single generating-function mechanism from which their properties follow.

What carries the argument

The load-bearing object is the degenerate exponential $e^x_{\lambda}(t)=(1+\lambda t)^{x/\lambda}$, along with the identity $e^{iy}_{\lambda}(t)=\cos^{(y)}_{\lambda}(t)+i\sin^{(y)}_{\lambda}(t)$, where the degenerate cosine and sine functions are $\cos^{(y)}_{\lambda}(t)=\cos((y/\lambda)\log(1+\lambda t))$ and $\sin^{(y)}_{\lambda}(t)=\sin((y/\lambda)\log(1+\lambda t))$. Multiplying the generating functions (4) and (5) by $e^x_{\lambda}(t)\cos^{(y)}_{\lambda}(t)$ and $e^x_{\lambda}(t)\sin^{(y)}_{\lambda}(t)$ defines the four new polynomial families as coefficient sequences. The derivations then expand the degenerate sine and cosine functions as power series in $\log(1+\lambda t)$, whose coefficients are Stirling numbers of the first kind, and use degenerate Stirling numbers of the second kind $S^{(2)}_{\lambda}(k,l)$ to factor $(e_{\lambda}(t)-1+1)^x$; these two Stirling expansions carry the explicit formulas.

What would settle it

Take a small case such as $n=2$, $\lambda=1$, $x=1$, $y=1$, expand the defining generating function for $E^{(c)}_{n,\lambda}(x,y)$ to order $t^2$, and compare the result with the double-sum formula in Theorem 2.3; any disagreement would show the expansion is wrong, while agreement supports the formal coefficient comparison.

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Extended reading notes

Core claim

In the paper's own terms, the central discovery is that for any nonzero real $\lambda$ the degenerate Euler polynomials $E_{n,\lambda}(x+iy)$ and degenerate Bernoulli polynomials $\beta_{n,\lambda}(x+iy)$ decompose as $$$E^{{(c)}}$_{n,\$\lambda$}(x,y)=\frac{E_{n,\$\lambda$}(x+iy)+E_{n,\$\lambda$}(x-iy)}{2},\qquad $E^{{(s)}}$_{n,\$\lambda$}(x,y)=\frac{E_{n,\$\lambda$}(x+iy)-E_{n,\$\lambda$}(x-iy)}{2i},$$ with the same pattern for $\beta$, where the left-hand sides are the new degenerate cosine/sine-Euler and cosine/sine-Bernoulli polynomials. The paper proves that these polynomials satisfy explicit double-sum expansions involving Stirling numbers of the first and second kind (Theorems 2.2, 2.3, 3.1), reflection identities $E^{(c)}_{n,\lambda}(1-x,y)=(-1)^nE^{(c)}_{n,-\lambda}(x,y)$ and $E^{(s)}_{n,\lambda}(1-x,y)=(-1)^{n+1}E^{(s)}_{n,-\lambda}(x,y)$ (Theorems 2.6, 3.2), and translation formulas in the variable $x$ (Proposition 2.5 and equations (57)-(58)). As $\lambda\to0$ these degenerate families reduce to the new type Euler polynomials of [15] and the corresponding Bernoulli analogues, which is exactly the affirmative answer the authors give to the reviewer's question.

Load-bearing premise

All identities are formal power-series identities in $t$, and the paper does not discuss convergence or analytic continuation; if comparing coefficients in this formal setting were not legitimate, the explicit formulas would not follow.

Editorial extensions

If this is right

The new type Euler and Bernoulli polynomial families are not independent constructions: they are the real and imaginary parts of the complex-variable classical polynomials, so every identity for the complex-variable families splits into a cosine identity and a sine identity.
Taking $\lambda\to0$ recovers the new type Euler polynomials of [15] and the corresponding new type Bernoulli polynomials, so the four degenerate families form a one-parameter deformation of those known families.
The explicit finite double sums involving Stirling numbers give a direct way to compute the coefficients of all four families to any finite order, without solving recurrences.
The reflection and translation formulas let arguments be shifted or reflected while only changing $\lambda$ to $-\lambda$ or introducing falling-factorial factors, giving symmetries separately for the real and imaginary components.

Reading between the lines

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

The same real/imaginary-part decomposition could be applied to other degenerate special functions, such as degenerate gamma functions or degenerate polylogarithms, to produce cosine and sine versions of those families, since the only requirement is a degenerate exponential with a complex argument.
Because the paper's identities are formal series identities, they should hold over any commutative ring where the coefficients make sense (for example, with $\lambda$ a nilpotent or a $p$-adic element), suggesting umbral or $\lambda$-adic interpretations beyond real $\lambda$.
The reflection symmetry swapping $\lambda$ and $-\lambda$ in Theorems 2.6 and 3.2 invites a numerical study of the joint zeros of $E^{(c)}_{n,\lambda}(x,y)$ and $E^{(s)}_{n,\lambda}(x,y)$ as functions of $\lambda$ and $y$; the sign flip $(-1)^{n+1}$ in the sine case indicates the two components may have different parity behavior.
The affirmative answer to the reviewer's question implies that any parametric variant of Euler or Bernoulli polynomials obtained by taking real and imaginary parts is essentially determined by the underlying complex-variable polynomial, so future 'new type' constructions can be framed as complex-variable statements from the outset.

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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

You can send this to a referee with confidence. The paper does what it says: it takes the degenerate Euler and Bernoulli polynomials of complex variable, splits off real and imaginary parts, and defines four new polynomial families — degenerate cosine/sine Euler and degenerate cosine/sine Bernoulli polynomials. The identities are standard generating-function manipulations: Stirling-number expansions, reflection symmetries, translation formulas, and a few convolution identities. Nothing here is deep, but almost all of it is correct, and the new definitions do not appear in the prior literature cited. The paper also gives a clean affirmative answer to the reviewer's question in MathSciNet about whether the new-type Euler polynomials can be recovered from complex-variable Euler polynomials by taking real and imaginary parts; the answer was known in the non-degenerate case, but the degenerate version is genuinely new.

The derivations are transparent. Theorems 2.1–2.6 and 3.1–3.4 check out. The Stirling expansions follow directly from the series for the degenerate exponential and the degenerate cosine/sine functions. The reflection identities are the usual lambda-to-minus-lambda tricks, and they are consistent. The formal power series manipulation is standard for this field; convergence is not discussed, but nobody in this literature expects analytic continuation, and the identities are valid as formal series.

Soft spots, in proportion: the misprint in Theorem 2.7 is real and needs fixing. The first display writes the inner binomial coefficient as (n choose l), but the derivation in equation (44) and the corresponding sine identity both use (n choose k). With (n choose l) the identity is false already at n=2 in the lambda->0 limit, while with (n choose k) it is true. So the statement, as printed, is wrong, but the proof shows the intended coefficient and the defect is local. There are also a few typos — 'sime-Euler' in the abstract, 'degenrate' in Section 3, a stray comma — that a copyedit would catch. The bibliography is heavy on the authors' own related work, which is a bit claustrophobic, but the central derivations do not depend on those papers; they are independent consequences of the definitions.

Who is this for? Specialists in degenerate special functions and umbral calculus, especially those working on polynomial families via generating functions. It is not a paper that will change anyone's research program. But it is a solid, citable note that answers a reviewer's question and adds several useful examples to the catalog. I would send it to peer review; the referee's job is light, mostly checking the typographical errors and confirming that the identities hold. I would not personally cite it unless I were actively working on degenerate trigonometric analogs, but for someone in that niche it deserves serious engagement.

Referee Report

1 major / 5 minor

Summary. The manuscript defines degenerate versions of the cosine and sine Euler and Bernoulli polynomials by substituting the degenerate exponential function into the standard generating functions and then separating real and imaginary parts of the complex variable. It derives explicit expansions involving degenerate Stirling numbers of the first and second kind, reflection symmetries with respect to x → 1−x and λ → −λ, translation identities, and relations between the degenerate cosine/sine polynomials and the new degenerate Euler and Bernoulli polynomials. The paper also claims to give an affirmative answer to a question posed by the reviewer of [15].

Significance. If correct, the paper provides a systematic and explicit degenerate analogue of the new type Euler and Bernoulli polynomials, with formulas that are directly verifiable by standard generating-function coefficient comparison. The derivations are transparent and the λ→0 limits are correctly stated. The novelty is moderate—most steps are formal manipulations of known generating functions—but the resulting formulas are explicit and the affirmative answer to the reviewer's question is clearly documented. The manuscript does not address convergence or analytic continuation; the identities should be understood as formal power-series identities.

major comments (1)

[Section 2, Theorem 2.7] The first display in Theorem 2.7 is false as printed: the inner binomial coefficient is written as (n choose l), but the derivation in Eq. (44) and the parallel sine identity in the same theorem both require (n choose k). With (n choose l), the identity already fails at n=2 in the λ→0 limit: the generating function 2/(e^t+1) e^{xt} cos(yt) gives E^c_2(x,y)=x^2-y^2-x, while the printed sum evaluates to x^2-y^2. The corrected version with (n choose k) evaluates correctly. Since Eq. (44) supplies the intended coefficient, this is a local typographical error rather than a gap in the derivation, but it must be corrected before publication.

minor comments (5)

[Abstract] The word 'sime-Euler' should be 'sine-Euler'.
[Section 3] The phrase 'degenrate sine-Bernoulli polynomials' appears in both the outline and the body of Section 3; it should be 'degenerate sine-Bernoulli polynomials'.
[Section 2, Theorem 2.1] In the second equality of Theorem 2.1, the notation E_{l,λ} is used without an explicit argument; writing E_{l,λ}(x) would remove ambiguity.
[Equation (46)] The left-hand side writes e^{x−iy}_λ without the argument (t); it should be e^{x−iy}_λ(t) for consistency with Eq. (45).
[Throughout] The paper should state explicitly that all generating-function identities are formal power series in t, so that convergence is not an issue; this would preempt a natural reader concern.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: every identity follows by coefficient comparison from the explicit generating-function definitions; self-citations concern standard background definitions only.

full rationale

The paper introduces degenerate cosine/sine Euler and Bernoulli polynomials directly via generating functions (24), (25), (49), (50), and all subsequent theorems (Theorems 2.1-2.7, 3.1-3.4, Proposition 2.5, Corollary 3.4) are obtained by expanding products of power series, using standard Stirling-number expansions, comparing coefficients, and elementary substitutions. No parameter is fitted, no 'prediction' is a renamed fit, and no external or self-cited result is invoked to force the central identities: the definitions of the degenerate exponential, degenerate Stirling numbers, and Carlitz polynomials are standard background, and the cited works by the present authors are used only for notation or standard facts (e.g., degenerate Stirling numbers of the second kind), not for the paper's novel conclusions. The complex-variable relations (26), (27), (51), (52) are immediate consequences of the generating functions and Euler's formula, not imported conclusions. One localized display issue exists in Theorem 2.7, where the first double sum contains an inner binomial coefficient written as (n choose l) while the proof in (44) and the companion sine identity require (n choose k); this makes the displayed formula false as printed in the λ→0 limit, but it is a typographical/correctness defect, not a circular step, because the proof supplies the intended coefficient and the derivation is independent. The paper is formally self-contained in its derivation chain, so the circularity score is 0.

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

The paper relies only on standard facts from special functions and formal power series; no ad hoc assumptions or fitted parameters are introduced. The parameter λ is an input variable, not fitted to data.

assumptions (4)

standard math Generating functions for ordinary Bernoulli and Euler polynomials exist and define polynomial sequences.
Used in (1), (2), (14), (15) as the starting point; standard definitions.
standard math The degenerate exponential e_λ(t) = (1+λt)^{1/λ} has the formal Taylor expansion sum (x)_{n,λ} t^n/n!.
Used throughout, e.g., in (28), (31)-(34).
standard math Stirling number expansions (6), (7), (8) hold as formal power series.
Used in Theorem 2.2, 2.3, 2.7, 3.1, etc.
standard math Euler's formula e^{ix}=cos x + i sin x and the real/imaginary part decomposition are valid.
Used to define cosine and sine polynomials in (16)-(25).

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Cite this review

Pith. "Pith review of A note on degenerate Euler and Bernoulli polynomials of complex variable." pith.science (2026). https://pith.science/paper/EGACCY3E

@misc{pith2026190803783,
  author       = {Pith},
  title        = {Pith review of: A note on degenerate Euler and Bernoulli polynomials of complex variable},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EGACCY3E}},
  note         = {Machine review of arXiv:1908.03783}
}

read the original abstract

In this paper, we study the degenerate version of the new type Euler polynomials, namely degenerate cosine-Euler polynomials and sime-Euler polynomials and also corresponding ones for Bernoulli polynomials, namely degenerate cosine Bernoulli polynomials and degenerate sine-Bernoulli polynomials by considering the degenerate Euler polynomials of complex variable and the degenerate Bernoulli polynomials of complex variable.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On type 2 degenerate Bernoulli and Euler polynomials of complex variable
math.NT 2019-08 conditional novelty 3.0 of 10

The paper introduces type 2 degenerate cosine and sine Bernoulli and Euler polynomials and derives explicit summation identities for them.

Reference graph

Works this paper leans on

20 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

Masjed-Jamei, M

M. Masjed-Jamei, M. R. Beyki, W. Koepf, A new type of Euler polynomials and numbers, Mediterr. J. Math. 15 (2018), no. 3, Art. 138, 17 pp. 16 DAE SAN KIM, TAEKYUN KIM, AND HYUNSEOK LEE

work page 2018

[2] [1]

Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math

L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51-88

1979

[3] [2]

Carlitz, A degenerate Staud-Clausen theorem, Arch

L. Carlitz, A degenerate Staud-Clausen theorem, Arch. Math. (Basel) 7 (1956), 28-33

work page 1956

[4] [3]

D. V . Dolgy, T. Kim, Some explicit formulas of degenerate Stirling numbers asso ciated with the degenerate special numbers and polynomials, Proc. Jangjeon Math. Soc. 21 (2018), no. 2, 309-317

2018

[5] [4]

Haroon, W

H. Haroon, W. A. Khan, Degenerate Bernoulli numbers and polynomials associated w ith degenerate Hermite poly- nomials, Commum Korean Math. Soc. 33 (2018), no. 2, 651-669

work page 2018

[6] [5]

G.-W. Jang, T. Kim, H.-I. Kwon, On the extension of degenerate Stirling polynomials of the s econd kind and degen- erate Bell polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 28 (2018), no. 2, 305-316

work page 2018

[7] [6]

D. S. Kim, T. Kim, A note on polyexponential and unipoly functions, Russ. J. Math. Phys. 26 (2019), no. 1, 40-49

work page 2019

[8] [7]

Kim, G.-W

T. Kim, G.-W. Jang, A note on degenerate gamma function and degenerate Stirling number of the second kind, Adv. Stud. Contemp. Math. (Kyungshang) 28 (2018), no. 2, 207-214

work page 2018