{"id":"ffedb311-e6ec-4a63-955f-a151793ce0b3","arxiv_id":"1908.03783","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces degenerate cosine-Euler, sine-Euler, cosine-Bernoulli, and sine-Bernoulli polynomials and proves several identities for them using generating functions.","lead":"This note defines degenerate cosine and sine Euler and Bernoulli polynomials by plugging a degenerate exponential into known generating functions. It derives explicit formulas, reflection symmetries, and translation identities for these new polynomial families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.7 contains a misprinted binomial coefficient; the central claim is sound once the display is corrected to match the proof in eq (44).","rationale":"The paper's formal-power-series framework is standard and the coefficient comparisons are valid: for fixed nonzero λ, each generating function has well-defined coefficients, and the rearrangements in (29)-(30), (33)-(34), (44), and (63) are finite coefficient-wise. I checked the reflection substitutions (42)-(43) and (55) including the sine sign, the translation formulas, and the Bernoulli difference identities; they are consistent. The single concrete defect I found is the misprinted binomial in the first display of Theorem 2.7, which contradicts the proof in eq (44). Because the reader's strongest claim cites the theorem statements verbatim, this warrants a required correction before acceptance; however, the proof supplies the correct identity and no other part of the central construction is affected, so conditional acceptance is the appropriate outcome rather than rejection.","tokens_in":12728,"tokens_out":40829,"duration_ms":358667,"concrete_test":"Set λ=0 in Theorem 2.7 and evaluate the n=2 case two ways: first compute E^c_2(x,y) directly from 2/(e^t+1)e^{xt}cos(yt), obtaining x^2-y^2-x; then evaluate the printed double sum with (n choose l), which gives x^2-y^2, and with (n choose k), which gives x^2-y^2-x. The mismatch confirms the required correction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The strongest claim explicitly includes the Stirling expansions in Theorems 2.1-2.7, and the first display of 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 fails already 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. This is a genuine internal inconsistency between the theorem statement and its proof, but because eq (44) supplies the intended coefficient, the underlying construction and all other identities appear valid; the defect is local and typographical rather than a gap in the central derivation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":12928,"tokens_out":3147,"duration_ms":31690,"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":[{"comment":"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.","section":"Section 2, Theorem 2.7"}],"minor_comments":[{"comment":"The word 'sime-Euler' should be 'sine-Euler'.","section":"Abstract"},{"comment":"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":"Section 3"},{"comment":"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.","section":"Section 2, Theorem 2.1"},{"comment":"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).","section":"Equation (46)"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a straightforward extension of the authors' previous work on degenerate polynomials, but it is internally sound aside from local typographical errors. The citation pattern is heavily self-referential, though this does not affect correctness. The manuscript fits the journal's scope in special functions and number theory, and the proposed revision is minor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSoft 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.\n\nWho 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.","headline":"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.","tokens_in":13389,"tokens_out":1267,"would_cite":false,"duration_ms":16654,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B68","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["degenerate cosine-Euler polynomials","degenerate sine-Euler polynomials","degenerate cosine-Bernoulli polynomials","degenerate sine-Bernoulli polynomials","degenerate cosine-polynomials","degenerate sine-polynomials","degenerate Stirling numbers of the second kind"],"falsifier":"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.","tokens_in":12570,"feed_emoji":"🧮","tokens_out":18264,"duration_ms":147560,"temperature":0.7,"pith_summary":"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.","feed_headline":"Complex variable splits degenerate Euler and Bernoulli polynomials","feed_subtitle":"Real and imaginary parts become cosine and sine polynomial families with explicit formulas.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the degenerate Bernoulli and Euler polynomials, the base objects into which the paper substitutes a complex variable.","marker":"[1,2]"},{"why":"Introduces the new type Euler polynomials and cosine/sine polynomials that the degenerate families generalize, and is the source of the reviewer's question.","marker":"[15]"},{"why":"Defines degenerate Stirling numbers of the second kind, used in the expansions of Theorems 2.7 and Section 3.","marker":"[9]"},{"why":"Supplies Stirling numbers of the first kind through the falling-factorial expansion, the coefficients appearing throughout the explicit double sums.","marker":"[7,11,17]"},{"why":"Provides the real/imaginary split of ordinary Euler and Bernoulli polynomials into cosine and sine polynomials, the pattern transferred to the degenerate setting.","marker":"[12]"}],"fun_headline_variants":["Degenerate complex polynomials split into cosine and sine","Cosine and sine arise from degenerate Euler and Bernoulli","Complex degenerate polynomials yield cosine and sine families","Splitting complex variable gives cos and sin polynomial pairs","Real and imaginary parts become degenerate cosine and sine"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Degenerate complex polynomials split into cosine and sine","Cosine and sine arise from degenerate Euler and Bernoulli","Complex degenerate polynomials yield cosine and sine families","Splitting complex variable gives cos and sin polynomial pairs","Real and imaginary parts become degenerate cosine and sine"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1375,"prompt_tokens":903,"completion_tokens":472,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":400}},"tokens_in":519,"tokens_out":472,"duration_ms":5318,"temperature":1.0,"reasoning_tokens":400,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:01:51.788454+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Masjed-Jamei, M","cited_arxiv_id":null,"evidence_quote":"Introduces the new type Euler polynomials and cosine/sine polynomials that the degenerate families generalize, and is the source of the reviewer's question."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the real/imaginary split of ordinary Euler and Bernoulli polynomials into cosine and sine polynomials, the pattern transferred to the degenerate setting."}],"review_version":1}