{"id":"e6cfe0bd-e5a7-461a-bd51-86858f7ad24c","arxiv_id":"1908.11009","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces type 2 degenerate cosine and sine Bernoulli and Euler polynomials and derives explicit summation identities for them.","lead":"This paper defines new families of polynomial functions by splitting type 2 degenerate Bernoulli and Euler polynomials of complex variable into real and imaginary parts. The main content is a set of explicit formulas and identities connecting these new cosine and sine polynomials to known polynomials and Stirling numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.8's displayed formula omits the reciprocal of the binomial coefficient; as printed it fails for n=1, k=1.","rationale":"The reader's weakest assumption concerned formal power series manipulation, which I find standard and non-load-bearing: all substitutions have zero constant term and unit linear term, and coefficient extraction is finite. My review instead identifies a concrete issue in Theorem 2.8: the displayed formula appears to drop the reciprocal of the binomial coefficient that is present in the derivation (2.31). For k=1, n=1, x=0 the printed identity fails numerically. This directly affects the central claim of explicit finite-sum identities. Since the derivation indicates the intended formula, the issue is likely typographical and does not invalidate the method; the paper should be conditionally accepted pending correction. The reader's overall CONDITIONAL verdict remains appropriate, but for a different, more specific reason.","tokens_in":85,"tokens_out":35829,"duration_ms":391527,"concrete_test":"Check the original PDF/LaTeX of Theorem 2.8: if it contains \\binom{n-j+k}{k}^{-1}, the concern is resolved; if not, evaluate both sides for k=1, n=1, x=0 (any y, λ): definition (2.25) gives -λ/2 while the printed RHS equals 1, disproving the identity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing issue is in (2.31) and Theorem 2.8. The derivation begins with the correct factor T_λ(l+k,k|x)/binom(l+k,k), but the displayed convolution and the theorem state binom(n-j+k,k) T_λ(n-j+k,k|x) without a reciprocal. If the printed formula is literal, Theorem 2.8 is false: for k=1, n=1, x=0, definition (2.25) gives B^{(c,-1)}_{1,λ}(0,y) = -λ/2 (the coefficient of t in (e^{1/2}_λ - e^{-1/2}_λ)/t · cos_yλ), whereas the printed sum gives 1 (the j=0 term). The intended identity requires division by binom(n-j+k,k). This is either a typo in the manuscript text or a genuine error; it is the one place where the explicit finite-sum claim can fail as stated. The other theorems (2.2, 2.5, 2.6, 2.9, 2.10) check out under formal power series manipulations.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12265,"tokens_out":8046,"duration_ms":70163,"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":[{"comment":"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.","section":"Section 2, equation (2.31) and Theorem 2.8"}],"minor_comments":[{"comment":"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.","section":"Equation (2.43)"},{"comment":"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.","section":"Equations (2.12) and Theorem 2.4"},{"comment":"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.","section":"Equation (2.15)"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Editorial"}],"recommendation":"major_revision","confidential_remarks":"The paper is a routine but potentially useful extension of the authors' earlier work. The main issue is the concrete algebraic error in Theorem 2.8 and equation (2.31), which is fixable but currently invalidates the stated negative-order formula. I would ask the authors to correct that error and re-check the other convolution identities for similar reciprocal errors before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a direct extension of two of the authors' own papers—ref [5] (type 2 degenerate Bernoulli/Euler polynomials) and ref [8] (complex-variable separation)—and the new results are exactly what you'd expect: new polynomial families plus finite-sum identities obtained by routine generating-function manipulations. The one thing to know before citing it: Theorem 2.8 as printed is false. The derivation in (2.31) correctly has T_λ(n-j+k,k|x)/(n-j+k choose k), but the theorem statement drops the reciprocal. For n=1, k=1, x=0, the formula gives -2λ instead of the correct -λ/2. This is a fixable typo, not a broken method, but it needs to be corrected and checked.\n\nWhat the paper does well: it defines the type 2 degenerate cosine/sine Bernoulli and Euler polynomials and their higher- and negative-order variants, and derives explicit expressions in terms of Stirling numbers and the base polynomials. The generating-function arguments are standard and, apart from Theorem 2.8, the coefficient extractions I spot-checked in (2.12), (2.14), (2.16), and (2.39) are consistent. Theorems 2.2, 2.5, 2.6, 2.7, 2.9, and 2.10 look right. The paper is honest about being a catalog-style contribution; it does not oversell the depth. There is no code or machine proof, but for formal power series identities of this kind none is expected.\n\nSoft spots, in proportion: (i) the Theorem 2.8 reciprocal error is load-bearing for that one theorem, but it is isolated and obvious once you check a small case; (ii) there is a minor index typo in (2.43) (E_{n,λ} should be E_{k,λ}); (iii) the abstract claims the real/imaginary-part idea gives an affirmative answer to Belbachir's question, but ref [8] already did that—this paper is the type 2 version, not the first answer; (iv) self-citation is heavy, but it is legitimate here because the base definitions genuinely come from the authors' own [5] and [8].\n\nWho is this for? Only specialists working on degenerate versions of special polynomials or umbral calculus identities. They will find the catalog useful and will need to repair Theorem 2.8. A serious referee should be able to check the rest in an afternoon.\n\nBottom line: worth a peer-review round, with mandatory correction of Theorem 2.8 and a pass over the other negative-order formulas. If the authors fix that, it is a publishable small contribution.","headline":"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.","tokens_in":12856,"tokens_out":4000,"would_cite":false,"duration_ms":34521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B83","05A19"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["type 2 degenerate Bernoulli polynomials of complex variable","type 2 degenerate Euler polynomials of complex variable","type 2 degenerate cosine-Bernoulli polynomials","type 2 degenerate sine-Bernoulli polynomials","type 2 degenerate cosine-Euler polynomials","type 2 degenerate sine-Euler polynomials","Stirling numbers of the first kind","Stirling numbers of the second kind"],"falsifier":"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.","tokens_in":11737,"feed_emoji":"🧮","tokens_out":7380,"duration_ms":71573,"temperature":0.7,"pith_summary":"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.","feed_headline":"Splitting complex parts yields new degenerate cosine-sine formulas","feed_subtitle":"Real and imaginary parts of complex-variable type 2 degenerate Bernoulli and Euler polynomials give explicit Stirling-number identities.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $\\lambda$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the defining identities (1.10) and (1.11) for the type 2 degenerate Bernoulli and Euler polynomials on which the whole construction is built.","marker":"[5]"},{"why":"Provides the earlier complex-variable construction of degenerate cosine and sine Bernoulli and Euler polynomials whose type 2 version is developed here.","marker":"[8]"},{"why":"Introduces the 'new type Euler polynomials' that the type 2 degenerate Euler polynomials are a degenerate version of.","marker":"[15]"},{"why":"Supplies the original degenerate Bernoulli polynomial framework, including the degenerate exponential functions used throughout.","marker":"[1, 2]"},{"why":"Defines the degenerate central factorial polynomials of the second kind used in the negative-order formula of Theorem 2.8.","marker":"[9]"},{"why":"Defines the central factorial numbers of the second kind used in the order-$\\alpha$ and negative-order computations.","marker":"[3]"}],"fun_headline_variants":["Real and imaginary parts produce new degenerate identities","Explicit Stirling sums for type 2 degenerate Bernoulli-Euler","Cosine-sine split yields degenerate polynomial formulas","New identities for type 2 degenerate Euler and Bernoulli"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Real and imaginary parts produce new degenerate identities","Explicit Stirling sums for type 2 degenerate Bernoulli-Euler","Cosine-sine split yields degenerate polynomial formulas","New identities for type 2 degenerate Euler and Bernoulli"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000331,"raw_usage":{"total_tokens":1828,"prompt_tokens":916,"completion_tokens":912,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":851}},"tokens_in":532,"tokens_out":912,"duration_ms":9262,"temperature":1.0,"reasoning_tokens":851,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:27:20.178055+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the defining identities (1.10) and (1.11) for the type 2 degenerate Bernoulli and Euler polynomials on which the whole construction is built."},{"cited_title":"A note on degenerate Euler and Bernoulli polynomials of complex variable","cited_arxiv_id":"1908.03783","evidence_quote":"Provides the earlier complex-variable construction of degenerate cosine and sine Bernoulli and Euler polynomials whose type 2 version is developed here."},{"cited_title":"Masjed-Jamei, M.R","cited_arxiv_id":null,"evidence_quote":"Introduces the 'new type Euler polynomials' that the type 2 degenerate Euler polynomials are a degenerate version of."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the central factorial numbers of the second kind used in the order-$\\alpha$ and negative-order computations."}],"review_version":1}