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On type 2 degenerate Bernoulli and Euler polynomials of complex variable

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Pith's one-line read The paper defines type 2 degenerate cosine- and sine-Bernoulli and Euler polynomials by splitting complex-variable type 2 degenerate Bernoulli and Euler polynomials into real and imaginary parts, and proves explicit finite-sum identities…

desk verdict Routine extension of the authors' earlier work; Theorem 2.8 is false as printed (missing reciprocal of a binomial coefficient), but the rest checks out and the error is easy to fix. read the letter →

arxiv 1908.11009 v1 pith:2WYKXC2P submitted 2019-08-29 math.NT

Taekyun Kim , Dae san Kim , Lee-Chae Jang , Han-Young Kim This is my paper · ORCID
classification math.NT MSC 11B8305A19
keywords type2degenerateBernoullipolynomialsofcomplexvariableEulercosine-Bernoullisine-Bernoullicosine-Eulersine-EulerStirlingnumbersthefirstkindsecond
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 introduces four new families of "type 2 degenerate" polynomials: cosine-Bernoulli, sine-Bernoulli, cosine-Euler, and sine-Euler, each in two variables $x$ and $y$. The construction starts from the type 2 degenerate Bernoulli and Euler polynomials of complex variable $x+iy$, averages them with the conjugates $x-iy$, and reads off the real and imaginary parts as separate generating functions. The central claim is that every such polynomial has an explicit finite-sum formula: a binomial convolution of an ordinary type 2 degenerate polynomial with powers of $y$ and $\lambda$ and Stirling numbers of the first kind. The paper also proves companion identities using Stirling numbers of the second kind, Bernoulli numbers of the second kind, and an order-$\alpha$ plus negative-order extension. This matters because it provides a general template for turning a complex-variable special polynomial into explicitly computable cosine and sine components, and it answers affirmatively a question raised in the literature about treating real and imaginary parts separately.

What carries the argument

The load-bearing machinery is the pair of generating functions for the four new families, built from the type 2 degenerate Bernoulli and Euler kernels $t/(e^{t/2}_\lambda-e^{-t/2}_\lambda)$ and $2/(e^{t/2}_\lambda+e^{-t/2}_\lambda)$ multiplied by $e^x_\lambda(t)$ and by the degenerate cosine and sine functions $\cos^{(y)}_\lambda(t)=\cos((y/\lambda)\log(1+\lambda t))$ and $\sin^{(y)}_\lambda(t)=\sin((y/\lambda)\log(1+\lambda t))$. The argument's central computational step expands $(\log(1+\lambda t))^{2m}$ and $(\log(1+\lambda t))^{2m+1}$ using Stirling numbers of the first kind, converting the product of the two generating functions into a finite binomial convolution. A second operation, substituting $t$ by $(1/\lambda)(e^{\lambda t}-1)$, swaps the first-kind Stirling expansion for a second-kind one and produces the complementary identities.

What would settle it

Pick $\lambda=1$, $x=0$, $y=1$, expand the generating function (2.8) directly through degree 2, and compare the coefficient of $t^2/2!$ with the formula for $B^{(s)}_{2,1}(0,1)$ given by Theorem 2.2; any disagreement would show that the interchange of summation or the coefficient extraction is invalid.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the type 2 degenerate cosine-Bernoulli polynomials $B^{(c)}_{n,\lambda}(x,y)$ and sine-Bernoulli polynomials $B^{(s)}_{n,\lambda}(x,y)$, defined by generating functions (2.7) and (2.8), satisfy the finite identities of Theorem 2.2, and the corresponding type 2 degenerate cosine-Euler and sine-Euler polynomials satisfy the analogous identities of Theorem 2.9. In each case the coefficient formula is a single binomial sum over $k$ and $m$ in which the type 2 degenerate polynomial appears at index $n-k$, the trigonometric variable contributes $(-1)^m y^{2m}$ or $y^{2m+1}$, and $\lambda$ enters as $\lambda^{k-2m}$ or $\lambda^{k-2m-1}$ multiplied by a Stirling number of the first kind. The same structural formula is proved at order $\alpha$ and at negative order, and a substitution identity expresses the type 2 degenerate Euler polynomial of complex variable in terms of the ordinary type 2 Euler polynomial and Stirling numbers of the second kind.

Load-bearing premise

The derivations treat all generating functions as formal power series and freely interchange the order of summation, a stance the paper never states explicitly; the formulas also rely, without re-proof, on the defining identities (1.10) and (1.11) from the cited prior work.

Editorial extensions

If this is right

  • Every type 2 degenerate cosine-Bernoulli and cosine-Euler polynomial is determined by finitely many values of the corresponding one-variable type 2 degenerate polynomials, so computing the one-variable family gives the two-variable one.
  • Setting $y=0$ in Theorem 2.2 yields $B^{(c)}_{n,\lambda}(x,0)=\beta_{n,\lambda}(x+\tfrac12)$, connecting the new cosine-Bernoulli polynomials to the earlier degenerate Bernoulli polynomials.
  • The negative-order theorem (Theorem 2.8) writes type 2 degenerate cosine-Bernoulli polynomials as finite sums of degenerate central factorial polynomials of the second kind, giving a concrete combinatorial model for those polynomials.
  • Theorem 2.10 gives an explicit change-of-basis identity: for each $n$, the ordinary type 2 Euler polynomial at $x+iy$ is a finite sum of $\lambda$-weighted Stirling numbers of the second kind times type 2 degenerate Euler polynomials at $x+iy$.
  • A companion identity records the same relation after splitting real and imaginary parts, so any computation with the type 2 degenerate cosine and sine Euler polynomials feeds back into ordinary type 2 Euler polynomials.

Reading between the lines

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

  • Implicit extension: the same real-and-imaginary split should produce type 2 degenerate tangent and secant families, since tangent and secant are rational combinations of sine and cosine; the coefficient formulas here give the starting point.
  • Implicit extension: because the identities are formal-power-series identities, they remain valid when $\lambda$ is any formal or nilpotent parameter, which may make them useful in umbral or operator calculus settings.
  • Testable extension: taking the limit $\lambda\to 0$ in Theorems 2.2 and 2.9 should recover the ordinary type 2 cosine and sine polynomial identities, and checking that the $\lambda$-dependence drops out cleanly is a direct sanity test of the formulas.
  • Implicit connection: the finite-sum structure suggests a computational shortcut for numerical evaluation, since each two-variable polynomial is a direct convolution involving only Stirling numbers, powers of $y$, and powers of $\lambda$.
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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 / 5 minor

Summary. The paper introduces several new families of polynomials: type 2 degenerate cosine- and sine-Bernoulli polynomials, type 2 degenerate cosine- and sine-Euler polynomials, their higher-order versions, and negative-order cosine-Bernoulli polynomials. These are defined by generating functions obtained from the type 2 degenerate Bernoulli and Euler polynomials of complex variable by separating real and imaginary parts. The main results are explicit finite-sum formulas expressing the new polynomials in terms of type 2 degenerate Bernoulli/Euler polynomials and Stirling numbers of the first or second kind, together with identities relating them to classical type 2 Bernoulli and Euler polynomials. The proofs are formal manipulations of generating functions, using the standard expansions of degenerate cosine and sine functions via Stirling numbers of the first kind.

Significance. If correct, the paper provides a systematic catalogue of explicit formulas for new two-variable polynomial families that extend previous work on degenerate versions of Bernoulli and Euler polynomials. The definitions are clear and the coefficient extractions are checkable by hand; the formal generating-function framework is standard. The results are not deep but could be useful as a reference for later applications. However, the negative-order theorem contains a concrete binomial-factor error, and since that theorem is one of the paper's principal claims, the current version cannot be accepted as stated.

major comments (1)
  1. [Section 2, equation (2.31) and Theorem 2.8] The displayed formula and the theorem state B^{(c,-k)}_{n,λ}(x,y) = Σ_j Σ_m binom(n,j) binom(n-j+k,k) T_λ(n-j+k,k|x) (-1)^m y^{2m} λ^{j-2m} S1(j,2m). This is incorrect: the binomial factor binom(n-j+k,k) should appear in the denominator, not the numerator. Equation (2.24) gives B^{(-k)}_{n,λ} = T_λ(n+k,k)/binom(n+k,k), so substituting this into the convolution that produces (2.31) yields a factor 1/binom(n-j+k,k), not binom(n-j+k,k). As printed, Theorem 2.8 fails; for example, with k=1, n=1, and x=0, the printed sum equals 2 T_λ(2,1|0) = -2λ, whereas the definition (2.25) directly gives B^{(c,-1)}_{1,λ}(0,y) = -λ/2. Replacing binom(n-j+k,k) by its reciprocal repairs the statement. This is a load-bearing error in one of the paper's central claims and must be corrected.
minor comments (5)
  1. [Equation (2.43)] In the intermediate line after the first equality, the factor should be (E_{k,λ}(x+iy)+E_{k,λ}(x-iy))/2, not (E_{n,λ}(x+iy)+E_{n,λ}(x-iy))/2; the index n is a typo for k. The final displayed identity appears otherwise correct.
  2. [Equations (2.12) and Theorem 2.4] The notation B_{l,λ}(x,y) inside the convolution is ambiguous and should be B^{(c)}_{l,λ}(x,y) (and similarly B^{(s)}_{l,λ}(x,y) in the second part of Theorem 2.4), since B_{l,λ}(x) already denotes the type 2 degenerate Bernoulli polynomials without trigonometric factors.
  3. [Equation (2.15)] The inner summation index l clashes with the outer summation index l; please rename one of them (for instance, use r for the inner sum) to avoid confusion.
  4. [Throughout] The paper should explicitly state that all generating functions are treated as formal power series over R or C. This would justify the substitutions t ↦ (e^{λt}-1)/λ and t ↦ (1/λ) log(1+λt) and the interchanges of summation without invoking convergence.
  5. [Editorial] There are a few typographical slips, including 'repsectively' in the paragraph after (2.24) and the backtick in 'Hac`ene' in the abstract. These should be corrected in revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; theorems are coefficient extractions from the defining generating functions, with only minor non-load-bearing self-citations.

full rationale

The paper defines B(c), B(s), E(c), E(s) and their order-alpha versions by generating functions (2.7)-(2.8), (2.25)-(2.26), (2.35)-(2.36), then derives finite-sum identities by expanding the degenerate cosine/sine functions through log(1+lambda t) and Stirling numbers of the first kind. For instance, Theorem 2.2 follows by equating coefficients in (2.7) with the product expansion (2.9), and Theorem 2.9 does the same for the Euler case. The substitutions in (2.14), (2.16), and (2.39) are standard formal-power-series changes of variable that re-express the same generating functions in terms of Stirling numbers of the second kind; Theorem 2.10 then follows directly. Theorem 2.8 uses the derived identity (2.24) together with the same expansion, and is not an assumption of its own conclusion. The cited earlier papers [5], [8], and [9] supply definitions and previously introduced polynomials/functions; those definitions are restated in the manuscript and are not load-bearing unexamined premises. No parameter is fitted, no uniqueness claim is imported from the authors' prior work, and no known result is merely renamed. I therefore find no circular step; the only substantive issue I noticed, a likely missing reciprocal binomial coefficient in the displayed formula of Theorem 2.8, is a correctness/typographical matter rather than a circularity of the derivation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 5 invented entities

The paper adds new definitions but no free parameters. It relies on previously established generating-function identities and standard formal-series manipulations. The new polynomial families are honestly defined and are the subject of study, not unexplained postulates.

assumptions (3)
  • standard math Formal power series setting: the generating functions are treated as formal series, so no convergence is required for the interchanges of sums in (2.9)-(2.10), (2.29)-(2.30).
    Invoked throughout, e.g., in (2.9) where the product of two exponential generating functions is rearranged. In formal power series over R or C, this is valid.
  • domain assumption The identities (1.10)-(1.11) defining type 2 degenerate Bernoulli and Euler polynomials, and (1.3), (1.6) for Stirling and degenerate exponentials, are correct as stated.
    These are the baselines from refs [5], [3], [9] on which all later theorems rely. If any baseline definition were mis-transcribed, Theorems 2.2-2.10 could fail.
  • standard math The central factorial numbers T_λ(n,k) in (1.9) satisfy the stated binomial identity (2.24).
    Used in Theorem 2.8; the identity follows from the generating function (1.9).
invented entities (5)
  • Type 2 degenerate cosine-Bernoulli polynomials B^c_{n,λ}(x,y)
    purpose: Object of study; defined by generating function (2.7) as the cosine part of the type 2 degenerate Bernoulli polynomial of complex variable.
    Explicitly defined by a generating function in (2.7); all theorems are identities about this definition. No external falsifiable prediction is made.
  • Type 2 degenerate sine-Bernoulli polynomials B^s_{n,λ}(x,y)
    purpose: Object of study; defined by generating function (2.8).
    Explicitly defined by a generating function in (2.8); identities are internal to the definition.
  • Type 2 degenerate cosine-Euler polynomials E^c_{n,λ}(x,y)
    purpose: Object of study; defined by generating function (2.35).
    Explicitly defined by a generating function in (2.35); no external evidence.
  • Type 2 degenerate sine-Euler polynomials E^s_{n,λ}(x,y)
    purpose: Object of study; defined by generating function (2.36).
    Explicitly defined by a generating function in (2.36); no external evidence.
  • Higher-order and negative-order variants B^{(c,α)}_{n,λ}(x,y), B^{(s,α)}_{n,λ}(x,y), B^{(c,-k)}_{n,λ}(x,y)
    purpose: Natural α-parameter generalizations of the cosine/sine Bernoulli polynomials, defined in (2.25)-(2.26).
    Defined by generating functions in (2.25)-(2.26); used in Theorems 2.7-2.8.

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Cite this review

Pith. "Pith review of On type 2 degenerate Bernoulli and Euler polynomials of complex variable." pith.science (2026). https://pith.science/paper/2WYKXC2P

@misc{pith2026190811009,
  author       = {Pith},
  title        = {Pith review of: On type 2 degenerate Bernoulli and Euler polynomials of complex variable},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WYKXC2P}},
  note         = {Machine review of arXiv:1908.11009}
}
read the original abstract

Recently, Masjed-Jamei-Beyki-Koepf studied the so called new type Euler polynomials without making use of Euler polynomials of complex variable. Here we study degenerate and type 2 versions of these new type Euler polynomials, namely the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials and also the corresponding ones for Bernoulli polynomials, namely the type 2 degenerate cosine-Bernoulli and type 2 degenerate sine-Bernoulli polynomials by considering the degenerate Euler and degenerate Bernoulli polynomials of complex variable and by treating the real and imaginary parts separately. We derive some explicit expressions for those new polynomials and some identities relating to them. Here we note that the idea of separating the real and imaginary parts separately gives an affirmative answer to the question asked by Belbachir.

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Reference graph

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