{"id":"0174e1e1-f247-405c-9892-4c7ca1d8edb9","arxiv_id":"2505.22500","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"This work introduces deformed bivariate q-Appell polynomials and derives many structural formulas, but a key characterization theorem contains an error that propagates into the examples.","lead":"The paper defines deformed bivariate q-Appell polynomials using a deformed q-exponential, then derives derivative formulas, operator representations, and Mehler and Rogers type identities. It generalizes several q-Appell families, including Bernoulli, Euler, and Genocchi analogues, but one central theorem is false as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8 is false as stated: its right-hand side depends on a while the left-hand side does not, and the n=1 coefficient comparison gives a1+a0 x+(2-a)a0 y for the right side, not P_1; the same erroneous identity is recycled for Bernoulli, Euler, and Genocchi polynomials.","rationale":"The paper's core construction is a generating-function definition, and several results such as Theorems 1-3, 6-7, and parts of Theorem 9 are direct coefficient extractions that are likely correct. The decisive defect is Theorem 8, which is presented as a structural characterization and is repeated verbatim in the three example families. The n=1 counterexample is unambiguous and shows the theorem cannot hold for generic a or α. This is an internal inconsistency, not a disagreement with external consensus, so it directly undermines the correctness of the central claims. The reader's stated weakest assumption pointed at unproved operator identities from [3]; that is a legitimate secondary gap and would justify requiring explicit verification or a precise reference, but the Theorem 8 failure is sufficient to reject the manuscript in its present form. I agree with the reader's overall REJECT verdict, so the recommendation is UNCHANGED. The manuscript also contains unfinished placeholders, such as the [?,?] citation and the sentence 'The proof of the following result is given' in Section 2.3, which reinforce that this is not yet a complete, reviewable paper; the mathematical counterexample remains the primary reason for rejection.","tokens_in":13410,"tokens_out":5835,"duration_ms":64563,"concrete_test":"Compute the coefficient of t/[1]_q! on both sides of Eq. (24) for n=1 and arbitrary α with a_0 ≠ 0, using the generating function (13) and the recursion (25). The left coefficient is a_1 + a_0(x+y); the right coefficient is a_1 + a_0(x+y) + (1-a)a_0 y. Since a_0, a_1, y are arbitrary, equality holds only for a=1 or a_0=0, so the identity fails as stated. To test the possible repair, redo the proof's factorization with eq(ayt,u) in place of the second eq(yt,u); this will show that the second factor in the product must be P^{(α)}_{n-k,q}(x, a y; u), not P^{(α)}_{n-k,q}(x,y;u).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is internal and elementary: Theorem 8, Eq. (24), claims P^{(α)}_{n,q}(x,y;u) = Σ_{k=0}^n [n; k]_q u^{C(k,2)} A_{k,q}(a;u) y^k P^{(α)}_{n-k,q}(x,y;u), with A_{k,q}(a;u) defined by Eq. (25). The left side is independent of the auxiliary parameter a, while the right side contains A_{k,q}(a;u) and hence depends on a. For n=1, the generating function (13) gives P_1 = a_1 + a_0(x+y), while Eq. (25) gives A_0 = 1 and A_1 = 1-a. Substituting into Eq. (24) yields a_1 + a_0(x+y) + (1-a)a_0 y, which equals P_1 only if a=1. The proof introduces eq(ayt,u) but then identifies the second factor with P^{(α)}_{n-k,q}(x,y;u) instead of shifting the y-argument by a; the correct identity would keep a in the argument, e.g. P^{(α)}_{n-k,q}(x, a y; u). This error is not isolated: the same identity is asserted for the deformed q-Bernoulli, q-Euler, and q-Genocchi polynomials in Eqs. (50), (65), and (80), so the advertised structural and addition properties in Section 4 collapse. Secondary to this, Theorems 4 and 5 inherit whatever status T(yD_q|u) and R_n have from [3]; if those properties are unproved, the operator-based results are conditional, but the Theorem 8 counterexample is already decisive on its own.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces deformed bivariate q-Appell polynomials P^{(α)}_{n,q}(x,y;u) via the generating function (A_q(t))^α e_q(tx) e_q(ty,u) = Σ P^{(α)}_{n,q}(x,y;u) t^n/[n]_q!, and derives a series of structural properties: explicit coefficient expansions (Theorems 1–3), q-derivative relations (Theorems 6–7), an operator representation using the deformed q-exponential operator T(yD_q|u) (Theorems 4–5), a binomial-type identity with an auxiliary parameter a (Theorem 8), addition formulas for the order parameter (Theorem 9), an algebraic group structure on the polynomial class (Theorems 11–12), Mehler and Rogers formulas for quasi-Appell polynomials (Theorems 16–17), and applications to deformed q-Bernoulli, q-Euler, and q-Genocchi polynomials (Section 4). Most displayed identities are formal power-series manipulations and appear correct, but two central structural identities—Theorem 8 and Theorem 9—are false as stated and are recycled in the examples.","tokens_in":13913,"tokens_out":7952,"duration_ms":84227,"significance":"If the remaining results were correct, the paper would provide a unified class of deformed bivariate q-Appell polynomials that specializes to known families (e.g., u=1, u=q) and yields explicit coefficient formulas, operator representations, and Mehler–Rogers identities. The paper's strength is its systematic formal manipulation and the explicit coefficient formulas, which are parameter-free in the deformation parameter. However, the false Theorem 8 and the inconsistent Theorem 9 undermine the advertised characterizations and the example identities, although both appear locally fixable. The operator-based Theorems 4–5 depend on unproved properties of T(yD_q|u) and R_n from the author's previous arXiv preprint [3], making those results conditional on an unpublished reference.","major_comments":[{"comment":"Theorem 8 is false as stated. For n=1, the generating function (13) gives P^{(α)}_{1,q}(x,y;u)=a_1+a_0(x+y), while the right-hand side of (24) with A_0=1 and A_1=1-a (from the displayed values after the proof) equals a_1+a_0(x+y)+(1-a)a_0 y. This matches only if a=1. The proof incorrectly identifies the second factor: after writing e_q(yt,u)/e_q(ayt,u) as the series with A_{k,q}(a;u), the remaining product is Σ P^{(α)}_{n,q}(x, a y; u) t^n/[n]_q!, not Σ P^{(α)}_{n,q}(x,y;u) t^n/[n]_q!. The correct identity should involve P^{(α)}_{n-k,q}(x, a y; u) on the right. This error propagates verbatim to Eqs. (50), (65), and (80) for the deformed q-Bernoulli, q-Euler, and q-Genocchi polynomials.","section":"§2.1, Theorem 8 (Eq. 24)"},{"comment":"The addition formula is internally inconsistent. The left-hand side uses deformation parameter u, while the right-hand side uses a different parameter v in P^{(β)}_{n-k,q}(y;v). The proof multiplies the generating functions for P^{(α)}(x) and P^{(β)}(y;v), which yields (A_q(t))^{α+β} e_q(tx) e_q(ty,v), not the left-hand side with e_q(ty,u). The identity is false for u≠v, and even for u=v the proof needs to be reconciled with the second variable. This error also affects Corollaries 1–2 and the 'addiction properties' Eqs. (57), (72), and (87).","section":"§2.1, Theorem 9 (Eq. 26)"},{"comment":"The operator representation and the operator action on generating functions rest entirely on the unproved properties T(yD_q|u){x^n}=R_n(x,y;u|q) and the linearity/action of T on infinite series, taken from the author's prior preprint [3]. Since [3] is cited as an arXiv preprint and is not included in this manuscript, the results are conditional on an external, not-yet-verified source. The paper should either prove these properties or explicitly state them as assumptions in the current work.","section":"§2.1, Theorems 4–5"}],"minor_comments":[{"comment":"In the induction step for the x-derivative, the final line incorrectly inserts the factor u^{(k+1 choose 2)}; this factor belongs only in the y-derivative case (Eq. 23). The statement of Eq. (22) is correct, but the proof as written is erroneous.","section":"§2.1, Theorem 7 proof"},{"comment":"There are several unresolved citation placeholders ('[?,?]') in §2.3, and the repeated phrase 'Addiction properties' should be 'Addition properties' in Eqs. (57), (72), and (87).","section":"§2.3 and Section 4"},{"comment":"Equation (39) contains a spurious 's' in e_q(x y t s, u); based on the derivation in the proof, it should be e_q(x y t, u).","section":"§3, Theorem 14 (Eq. 39)"},{"comment":"In the definition of A^{β}_{q,k}(t), the coefficient is written as a^{(α)}_{n+k}; it should be a^{(β)}_{n+k} to match the superscript of the polynomial sequence being used.","section":"§3, Theorem 16 (Eq. 42)"},{"comment":"The use of 'py' in expressions such as P^{(β)}_{n,q}(py;v) is ambiguous between the product p·y and the variable y; a distinct letter or explicit spacing would improve readability.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The paper's central program is salvageable, but the current version contains two false structural identities (Theorem 8 and Theorem 9) that are repeated in the examples. The author should carefully re-derive these identities, ideally with numerical checks for small n, and should either prove the needed operator properties from [3] or clearly mark them as imported assumptions. The paper would also benefit from a thorough proofreading pass to fix the numerous typos and unresolved citations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my quick read of Orozco López, arXiv:2505.22500.\n\nThe paper introduces a bivariate extension of the deformed q-Appell polynomials via the generating function (A_q(t))^α e_q(tx) e_q(ty,u), and derives the usual portfolio: explicit coefficients, q-derivative relations, operator representations, and Mehler/Rogers formulas. That is a natural definition, and the coefficient formulas in Theorems 2 and 3 are correct and straightforward. The operator formalism is lifted from the author's earlier paper [3]; if that paper's facts are solid, Theorems 4 and 5 are fine conditional statements. The Mehler/Rogers derivations are standard Leibniz-rule manipulations and, on a quick scan, appear formally correct.\n\nThe problem is Theorem 8, and it is not minor. Equation (24) claims P_n = Σ_k [n;k]_q u^{C(k,2)} A_k(a;u) y^k P_{n-k}(x,y;u). The left side does not depend on the auxiliary parameter a; the right side does. For n=1, Eq. (13) gives P_1 = a_1 + a_0(x+y), while A_0=1 and A_1=1-a from Eq. (25). Substituting into (24) yields P_1 + (1-a)a_0 y, which agrees only when a=1. The proof inserts eq(ayt,u) and then misidentifies the second factor: the y-argument should be scaled by a, e.g. P_{n-k}(x, a y; u). The same false identity is recycled for the Bernoulli, Euler, and Genocchi polynomials in (50), (65), and (80), so the advertised structural properties in Section 4 collapse.\n\nThere is also a smaller proof bug in Theorem 7: the x-derivative induction acquires a spurious u^{(k+1 choose 2)} factor in the displayed line. And the manuscript has unfinished remnants ('[?,?]' in Section 2.3, 'addiction properties') that should have been cleaned before submission. None of these are fatal in themselves, but they add noise.\n\nAll that said, the paper is not nonsense. The class is genuinely new relative to the cited literature, and most of the generating-function algebra appears correct. The error in Theorem 8 is local and fixable: correct the argument shift and re-derive the example sections. If the author does that, the paper is a reasonable contribution to q-special function theory.\n\nMy recommendation: do not accept as-is; send it to a referee anyway, because the construction is coherent enough that a specialist can verify whether the corrected identity and the Mehler/Rogers formulas hold. A desk rejection would waste a potentially salvageable paper. For a reading group, I would not bother unless someone wants to sharpen their eye for false parameter identities.","headline":"New bivariate deformed q-Appell class, but the central identity is false and its consequences collapse.","tokens_in":14400,"tokens_out":5051,"would_cite":false,"duration_ms":56316,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A30","11B83","11B68"],"pacs":[],"model":"deepseek-v4-flash","headline":"A one-parameter deformation of the bivariate q-Appell generating function creates a unified polynomial class with explicit formulas, operator representations, and Mehler–Rogers identities, covering Bernoulli, Euler, and Genocchi families.","keywords":["q-Appell polynomials","deformed bivariate q-Appell polynomials","deformed q-exponential function","deformed homogeneous polynomials","deformed q-Bernoulli polynomials","deformed q-Euler polynomials","deformed q-Genocchi polynomials","Mehler and Rogers formulas"],"falsifier":"Set $\\alpha=1$, $A_q(t)=1+t$, and $u=q$ in Theorem 4. The $t^2$ coefficient of $(1+t)e_q(tx)e_q(ty,q)$ must equal $T(yD_q|q)\\{x^2+[2]_q x\\}=R_2(x,y;q|q)+[2]_qR_1(x,y;q|q)$; a direct expansion of the left side gives $x^2+[2]_qxy+qy^2+[2]_qx+[2]_qy$. Computing $R_1$ and $R_2$ from the definition of the deformed homogeneous polynomials and checking this identity for a few values of $q$ would settle the operator representation.","tokens_in":13217,"feed_emoji":"🧮","tokens_out":16439,"duration_ms":144357,"temperature":0.7,"pith_summary":"The paper introduces bivariate polynomial sets of deformed q-Appell type by putting the u-deformed q-exponential in the second factor of the generating function, $(A_q(t))^\\alpha e_q(tx)e_q(ty,u)$. It claims that this one-parameter change yields the full Appell machinery in two variables: explicit q-binomial coefficient formulas, q-derivative rules in $x$ and $y$, an operator representation through the deformed q-exponential operator $T(yD_q|u)$, addition formulas in the order $\\alpha$, and Mehler- and Rogers-type identities for quasi-q-Appell polynomials. The same class is closed under a convolution product that makes it a commutative group. If correct, the construction unifies the deformed q-Bernoulli, q-Euler, and q-Genocchi families and specializes to earlier q-Appell and $(p,q)$-Appell classes when $u$ is chosen appropriately.","feed_headline":"One deformation parameter unifies bivariate q-Appell families","feed_subtitle":"Bernoulli, Euler, and Genocchi cases all emerge from the same deformed generating function.","key_machinery":"The central object is the deformed q-exponential $e_q(z,u)=\\sum_{n\\ge0}u^{\\binom n2}z^n/[n]_q!$, which replaces the second q-exponential factor in the defining identity $(A_q(t))^\\alpha e_q(tx)e_q(ty,u)=\\sum_n P_{n,q}^{(\\alpha)}(x,y;u)t^n/[n]_q!$. The $u^{\\binom n2}$ factor is the deformation mechanism: it makes the $y$-derivative produce a factor $u$, and it connects the class to the deformed homogeneous polynomials $R_n(x,y;u|q)$ through the deformed q-exponential operator $T(yD_q|u)$, defined on monomials by $T(yD_q|u)\\{x^n\\}=R_n(x,y;u|q)$. The q-Leibniz rule then carries the operator through products and produces the Mehler and Rogers identities.","core_discovery":"The paper's central claim is that the generating function $(A_q(t))^\\alpha e_q(tx)e_q(ty,u)=\\sum_n P_{n,q}^{(\\alpha)}(x,y;u)t^n/[n]_q!$ defines a genuine bivariate q-Appell class. The polynomials have explicit expansions in deformed homogeneous polynomials $R_n(x,y;u|q)$ and in the univariate deformed q-Appell polynomials; they obey the q-derivative rules $D_{q,x}P_{n,q}^{(\\alpha)}=[n]_qP_{n-1,q}^{(\\alpha)}(x,y;u)$ and $D_{q,y}P_{n,q}^{(\\alpha)}=[n]_qP_{n-1,q}^{(\\alpha)}(x,uy;u)$; and they are obtained from the univariate case by the operator $T(yD_q|u)$. The paper also develops a convolution algebra in which the deformed q-Appell sets form a commutative group, derives addition formulas in the order $\\alpha$ (including an expression of $R_n$ as a convolution of $P^{(\\alpha)}$ with $P^{(-\\alpha)}$), and proves Mehler- and Rogers-type identities for the quasi-q-Appell polynomials. The deformed q-Bernoulli, q-Euler, and q-Genocchi families are then exhibited as instances of the construction.","pith_inferences":["A symmetric two-parameter version with $e_q(tx,u_1)e_q(ty,u_2)$ is the natural next step; the coefficient formulas in the paper show exactly where a second weight $u_2^{\\binom m2}$ would enter, so the same methods should go through.","The $u^{\\binom n2}$ weights are the standard weights in partition enumeration, so the explicit coefficients are natural candidates for combinatorial models such as weighted lattice paths or partition statistics; the paper does not attempt this.","Because the convolution group structure allows reciprocals of determining functions, applying it to the Bernoulli and Genocchi generating functions would produce explicit inversion identities not stated in the examples.","The Mehler and Rogers proofs are formal power series computations; extending them to convergence statements would require analytic assumptions on $A_q$ and $T(yD_q|u)$ that the paper leaves implicit."],"forward_implications":["For every fixed $u$, the polynomials satisfy the two annihilation rules $D_{q,x}P_{n,q}^{(\\alpha)}=[n]_qP_{n-1,q}^{(\\alpha)}(x,y;u)$ and $D_{q,y}P_{n,q}^{(\\alpha)}=[n]_qP_{n-1,q}^{(\\alpha)}(x,uy;u)$, so the defining Appell property transfers to both variables.","Theorem 3 and Corollary 2 together express the deformed homogeneous polynomials $R_n(x,y;u|q)$ as convolutions of positive- and negative-order deformed q-Appell polynomials, giving a change of basis between two natural polynomial bases.","The convolution operation makes the deformed q-Appell class a commutative group, so products and inverses of determining functions stay inside the class and produce new polynomial sets from old ones.","The Mehler and Rogers identities give closed generating functions for infinite sums of products of quasi-q-Appell and deformed q-Appell polynomials, and in the examples they become explicit identities for deformed q-Bernoulli, q-Euler, and q-Genocchi polynomials."],"supporting_citations":[{"why":"It defines the classical Appell polynomials whose generating function is the template being generalized.","marker":"[1]"},{"why":"It introduces q-Appell polynomials of type I, the univariate q-model that this construction generalizes.","marker":"[2]"},{"why":"It supplies the deformed homogeneous polynomials $R_n$ and the deformed q-exponential operator $T(yD_q|u)$ used in Theorems 3 through 5.","marker":"[3]"},{"why":"It provides q-Appell polynomials of type II, another q-analogue subsumed by the deformed class for a suitable choice of $u$.","marker":"[4]"},{"why":"It gives the $(p,q)$-Appell polynomials, a two-parameter family that the $u$-deformed class generalizes.","marker":"[5]"}],"fun_headline_variants":["Bivariate q-Appell: one deformed parameter, three classic families","Deformed q-Appell polynomials enter two dimensions","Unified q-Bernoulli, Euler, Genocchi via deformed bivariate Appell","New operator yields Mehler and Rogers formulas for q-Appell sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the deformed q-exponential operator $T(yD_q|u)$ from the author's earlier work satisfies $T(yD_q|u)\\{x^n\\}=R_n(x,y;u|q)$ and can be interchanged with infinite sums; if either fails, the operator representation and the Mehler and Rogers derivations do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Bivariate q-Appell: one deformed parameter, three classic families","Deformed q-Appell polynomials enter two dimensions","Unified q-Bernoulli, Euler, Genocchi via deformed bivariate Appell","New operator yields Mehler and Rogers formulas for q-Appell sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3019,"prompt_tokens":911,"completion_tokens":2108,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":2032}},"tokens_in":527,"tokens_out":2108,"duration_ms":16076,"temperature":1.0,"reasoning_tokens":2032,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:06:48.884678+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $\\alpha=1$, $A_q(t)=1+t$, and $u=q$ in Theorem 4. The $t^2$ coefficient of $(1+t)e_q(tx)e_q(ty,q)$ must equal $T(yD_q|q)\\{x^2+[2]_q x\\}=R_2(x,y;q|q)+[2]_qR_1(x,y;q|q)$; a direct expansion of the left side gives $x^2+[2]_qxy+qy^2+[2]_qx+[2]_qy$. Computing $R_1$ and $R_2$ from the definition of the deformed homogeneous polynomials and checking this identity for a few values of $q$ would settle the operator representation.","supporting_citations":[{"cited_title":"Appell, Sur une classe de polynomes, Ann","cited_arxiv_id":null,"evidence_quote":"It defines the classical Appell polynomials whose generating function is the template being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces q-Appell polynomials of type I, the univariate q-model that this construction generalizes."},{"cited_title":"Deformed Newton's $(s,t)$-Binomial Series and Generating Functions of Generalized Central Binomial Coefficients and Generalized Catalan Numbers","cited_arxiv_id":"2306.07431","evidence_quote":"It supplies the deformed homogeneous polynomials $R_n$ and the deformed q-exponential operator $T(yD_q|u)$ used in Theorems 3 through 5."},{"cited_title":"On a new $q$-analogue of Appell polynomials","cited_arxiv_id":"1801.08859","evidence_quote":"It provides q-Appell polynomials of type II, another q-analogue subsumed by the deformed class for a suitable choice of $u$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the $(p,q)$-Appell polynomials, a two-parameter family that the $u$-deformed class generalizes."}],"review_version":1}