Deformed Bivariate $q$-Appell Polynomials

Ronald Orozco L\'opez

REVIEW 3 major objections 5 minor 5 references

Deformed Bivariate $q$-Appell Polynomials

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict New bivariate deformed q-Appell class, but the central identity is false and its consequences collapse. read the letter →

arxiv 2505.22500 v1 pith:IF6CIU3Q submitted 2025-05-28 math.CO

Ronald Orozco López This is my paper

classification math.CO MSC 05A3011B8311B68

keywords q-Appellpolynomialsdeformedbivariateq-exponentialfunctionhomogeneousq-Bernoulliq-Eulerq-GenocchiMehlerandRogersformulas

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Here is my quick read of Orozco López, arXiv:2505.22500.

The 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.

The 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.

There 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.

All 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.

My 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§2.1, Theorem 8 (Eq. 24)] 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.
[§2.1, Theorem 9 (Eq. 26)] 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).
[§2.1, Theorems 4–5] 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.

minor comments (5)

[§2.1, Theorem 7 proof] 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.
[§2.3 and Section 4] 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).
[§3, Theorem 14 (Eq. 39)] 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).
[§3, Theorem 16 (Eq. 42)] 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.
[Notation] 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.

Circularity Check

1 steps flagged · score 2.0 of 10

No fitted-input circularity; only a minor definitional/self-citational operator reformulation.

self definitional [Theorem 4, Eq. (18), Section 2.1]
"Ppα q n,q px, y ; uq “ TpyDq|uqtPpα q n,q pxqu (18) where TpyDq|uqt xnu “ Rnpx, y ; u|qq (see [3])."

The operator T is invoked through its defining action on monomials, T{x^n}=R_n(x,y;u|q), and R_n is the coefficient of e_q(tx)e_q(ty,u) from the author's prior paper [3]. Theorem 3 already gives P^{(α)}_{n,q}(x,y;u)=Σ [n;k]_q a^{(α)}_k R_{n-k}(x,y;u|q). Applying the linear operator T to P^{(α)}_{n,q}(x)=Σ [n;k]_q a^{(α)}_k x^{n-k} and using T{x^{n-k}}=R_{n-k} reproduces Eq. (18) verbatim. Thus the operator representation is a notational restatement of Theorem 3 under the defining action of T, with the cited [3] property as the only external input. No parameter is fitted, so this is a mild definitional circularity rather than a forced or empirical prediction.

full rationale

The paper's derivation chain is largely self-contained: Definitions 1 and 2 fix the deformed bivariate q-Appell polynomials by an explicit generating function, and Theorems 1–3, 6–7, and 9–17 are direct coefficient manipulations of formal q-exponential series with no fitted parameters. The deformation parameter u and order α are formal variables, not data-derived constants. The one self-citational load-bearing point is the deformed q-exponential operator T(yD_q|u) and the polynomials R_n from the author's prior paper [3]; because T is used through its defining action T{x^n}=R_n, Theorems 4 and 5 reduce by construction to Theorem 3 and to the cited definition, respectively. This is a genuine but minor definitional/self-citational step, not a fitted input called a prediction. The skeptical objection to Theorem 8 — that its right-hand side depends on the auxiliary parameter a while the left-hand side does not — is a mathematical correctness issue in the statement or proof, not a circularity of the kind assessed here. Overall, the central explicit formulas and structural identities retain independent content, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The paper introduces no fitted numeric parameters; u and α are formal deformation and order parameters in the defining generating function. The main unproved inputs are the deformed q-exponential and the operator T(yD_q|u), taken from the author's prior arXiv paper [3], plus standard formal power series and q-calculus facts. The invented entities are new polynomial classes and operators defined for the purpose of extending q-Appell theory.

assumptions (4)

domain assumption The deformed q-exponential e_q(z,u) from [3] is a well-defined formal power series for all complex u, with the stated expansion.
Invoked in Definition 1 (Eq 10) and throughout the paper; the properties of this series are taken from the author's prior work.
domain assumption The deformed homogeneous polynomials R_n(x,y;u|q) and the operator T(yD_q|u) from [3] satisfy T(yD_q|u){x^n} = R_n(x,y;u|q) and are linear.
Used in Theorems 4 and 5 and in the Bernoulli, Euler, and Genocchi sections; the paper does not reprove these properties.
standard math The Leibniz rule for D_q (Eq 2) and the standard q-binomial coefficient identities hold.
Explicitly stated in Section 1 and used in the proofs of Theorems 16 and 17.
standard math Formal power series in x, y, t over C, with coefficients involving q, form an integral domain and allow division by series with nonzero constant term.
Needed for the generating function manipulations, particularly the ratio e_q(yt,u)/e_q(ayt,u) in Theorem 8.

invented entities (3)

Deformed bivariate q-Appell polynomials P^{(α)}_{n,q}(x,y;u)
purpose: New polynomial family generalizing univariate deformed q-Appell polynomials to two variables.
Defined by generating function (13); no external falsifiable handle is given beyond the defining formula.
Deformed q-Appell operators A^α(yD_q|u) and A^α(x,y;D_q|u)
purpose: Operator representations for the new polynomials and for quasi-q-Appell polynomials.
Defined in Section 3 as infinite sums of D_q powers; they are formal constructions internal to the paper.
Quasi-q-Appell polynomials Q^{(α)}_{n,q}(x,y;u) and Q^{(α)}_{n,q}(x,y,z;u)
purpose: Intermediate objects used to derive Mehler and Rogers type formulas.
Defined in Section 3 by explicit finite sums; no independent evidence outside the paper.

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Cite this review

Pith. "Pith review of Deformed Bivariate $q$-Appell Polynomials." pith.science (2026). https://pith.science/paper/IF6CIU3Q

@misc{pith2026250522500,
  author       = {Pith},
  title        = {Pith review of: Deformed Bivariate $q$-Appell Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IF6CIU3Q}},
  note         = {Machine review of arXiv:2505.22500}
}

abstract

In this paper, we introduce bivariate polynomial sets of deformed $q$-Appell type, and we study the algebraic properties of these sets. We show the relation between deformed bivariate $q$-Appell polynomials and deformed homogeneous polynomials. Next, we give some of their characterizations and algebraic structure. Then, we introduce the deformed $q$-Appell operators and obtain Mehler's and Rogers-type formulas of quasi-$q$-Appell polynomials. Finally, some examples of polynomial sequences of deformed $q$-Appell type are given: Bernoulli, Euler, and Genocchi types.

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Works this paper leans on

5 extracted references · 5 canonical work pages

[3]

Deformed Newton's $(s,t)$-Binomial Series and Generating Functions of Generalized Central Binomial Coefficients and Generalized Catalan Numbers

R. Orozco, Deformed homogeneous polynomials and the deforme d q-exponential operator, arXiv:2306.07431v4, (2024)

work page Pith review arXiv 2024
[1]

Appell, Sur une classe de polynomes, Ann

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[2]

W. A. Al-Salam, q-Appell polynomials, Ann. Mat. Pura Appl. 77 (1967), 31–45

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P. N. Sadjang, On new q-analogue of Appell polynomials, arXiv:1801.08859v1, (2018)

work page Pith review arXiv 2018
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P. N. Sadjang, On pp, q q-Appell polynomials, Anal. Math. 45 (2019) 583–598. 19

work page 2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

Deformed Newton's $(s,t)$-Binomial Series and Generating Functions of Generalized Central Binomial Coefficients and Generalized Catalan Numbers

R. Orozco, Deformed homogeneous polynomials and the deforme d q-exponential operator, arXiv:2306.07431v4, (2024)

work page Pith review arXiv 2024

[2] [1]

Appell, Sur une classe de polynomes, Ann

P. Appell, Sur une classe de polynomes, Ann. Sci. Ec. Norm. Sup´ er. 9 (1880) 119–144

work page

[3] [2]

W. A. Al-Salam, q-Appell polynomials, Ann. Mat. Pura Appl. 77 (1967), 31–45

work page 1967

[4] [4]

P. N. Sadjang, On new q-analogue of Appell polynomials, arXiv:1801.08859v1, (2018)

work page Pith review arXiv 2018

[5] [5]

P. N. Sadjang, On pp, q q-Appell polynomials, Anal. Math. 45 (2019) 583–598. 19

work page 2019